Precalculus Examples

$\frac{x^{4} + x^{2} + 1}{x^{2} {(2 x^{2} + 3)}^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A}{x^{2}} + \frac{B}{x} + \frac{C x + D}{{(2 x^{2} + 3)}^{2}}$

Step 1.2

$\frac{A}{x^{2}} + \frac{B}{x} + \frac{C x + D}{{(2 x^{2} + 3)}^{2}} + \frac{F x + G}{2 x^{2} + 3}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x^{2} {(2 x^{2} + 3)}^{2}$ .

$\frac{(x^{4} + x^{2} + 1) (x^{2} {(2 x^{2} + 3)}^{2})}{x^{2} {(2 x^{2} + 3)}^{2}} = \frac{A (x^{2} {(2 x^{2} + 3)}^{2})}{x^{2}} + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.4

Cancel the common factor of $x^{2}$ .

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Step 1.4.1

Cancel the common factor.

Step 1.4.2

Rewrite the expression.

$\frac{(x^{4} + x^{2} + 1) ({(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} = \frac{A (x^{2} {(2 x^{2} + 3)}^{2})}{x^{2}} + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.5

Cancel the common factor of ${(2 x^{2} + 3)}^{2}$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(x^{4} + x^{2} + 1) {(2 x^{2} + 3)}^{2}}{{(2 x^{2} + 3)}^{2}} = \frac{A (x^{2} {(2 x^{2} + 3)}^{2})}{x^{2}} + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.5.2

Divide $x^{4} + x^{2} + 1$ by $1$ .

$x^{4} + x^{2} + 1 = \frac{A (x^{2} {(2 x^{2} + 3)}^{2})}{x^{2}} + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x^{2}$ .

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Step 1.6.1.1

Cancel the common factor.

Step 1.6.1.2

Divide $A ({(2 x^{2} + 3)}^{2})$ by $1$ .

$x^{4} + x^{2} + 1 = A ({(2 x^{2} + 3)}^{2}) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.2

Rewrite ${(2 x^{2} + 3)}^{2}$ as $(2 x^{2} + 3) (2 x^{2} + 3)$ .

$x^{4} + x^{2} + 1 = A ((2 x^{2} + 3) (2 x^{2} + 3)) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.3

Expand $(2 x^{2} + 3) (2 x^{2} + 3)$ using the FOIL Method.

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Step 1.6.3.1

Apply the distributive property.

$x^{4} + x^{2} + 1 = A (2 x^{2} (2 x^{2} + 3) + 3 (2 x^{2} + 3)) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.3.2

Apply the distributive property.

$x^{4} + x^{2} + 1 = A (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2} + 3)) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.3.3

Apply the distributive property.

$x^{4} + x^{2} + 1 = A (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4

Simplify and combine like terms.

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Step 1.6.4.1

Simplify each term.

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Step 1.6.4.1.1

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = A (2 \cdot (2 x^{2} x^{2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.6.4.1.2.1

Move $x^{2}$ .

$x^{4} + x^{2} + 1 = A (2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = A (2 \cdot (2 x^{2 + 2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.2.3

Add $2$ and $2$ .

$x^{4} + x^{2} + 1 = A (2 \cdot (2 x^{4}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.3

Multiply $2$ by $2$ .

$x^{4} + x^{2} + 1 = A (4 x^{4} + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.4

Multiply $3$ by $2$ .

$x^{4} + x^{2} + 1 = A (4 x^{4} + 6 x^{2} + 3 (2 x^{2}) + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.5

Multiply $2$ by $3$ .

$x^{4} + x^{2} + 1 = A (4 x^{4} + 6 x^{2} + 6 x^{2} + 3 \cdot 3) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.1.6

Multiply $3$ by $3$ .

$x^{4} + x^{2} + 1 = A (4 x^{4} + 6 x^{2} + 6 x^{2} + 9) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.4.2

Add $6 x^{2}$ and $6 x^{2}$ .

$x^{4} + x^{2} + 1 = A (4 x^{4} + 12 x^{2} + 9) + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.5

Apply the distributive property.

$x^{4} + x^{2} + 1 = A (4 x^{4}) + A (12 x^{2}) + A \cdot 9 + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.6

Simplify.

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Step 1.6.6.1

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + A (12 x^{2}) + A \cdot 9 + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.6.2

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + A \cdot 9 + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.6.3

Move $9$ to the left of $A$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 \cdot A + \frac{B (x^{2} {(2 x^{2} + 3)}^{2})}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.7

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.6.7.1

Factor $x$ out of $B (x^{2} {(2 x^{2} + 3)}^{2})$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + \frac{x (B (x {(2 x^{2} + 3)}^{2}))}{x} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.7.2

Cancel the common factors.

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Step 1.6.7.2.1

Raise $x$ to the power of $1$ .

Step 1.6.7.2.2

Factor $x$ out of $x^{1}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + \frac{x (B (x {(2 x^{2} + 3)}^{2}))}{x \cdot 1} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.7.2.3

Cancel the common factor.

Step 1.6.7.2.4

Rewrite the expression.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + \frac{B (x {(2 x^{2} + 3)}^{2})}{1} + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.7.2.5

Divide $B (x {(2 x^{2} + 3)}^{2})$ by $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x {(2 x^{2} + 3)}^{2}) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.8

Rewrite ${(2 x^{2} + 3)}^{2}$ as $(2 x^{2} + 3) (2 x^{2} + 3)$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x ((2 x^{2} + 3) (2 x^{2} + 3))) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.9

Expand $(2 x^{2} + 3) (2 x^{2} + 3)$ using the FOIL Method.

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Step 1.6.9.1

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 x^{2} (2 x^{2} + 3) + 3 (2 x^{2} + 3))) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.9.2

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2} + 3))) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.9.3

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10

Simplify and combine like terms.

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Step 1.6.10.1

Simplify each term.

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Step 1.6.10.1.1

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 \cdot (2 x^{2} x^{2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.6.10.1.2.1

Move $x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 \cdot (2 x^{2 + 2}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.2.3

Add $2$ and $2$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (2 \cdot (2 x^{4}) + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.3

Multiply $2$ by $2$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (4 x^{4} + 2 x^{2} \cdot 3 + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.4

Multiply $3$ by $2$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (4 x^{4} + 6 x^{2} + 3 (2 x^{2}) + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.5

Multiply $2$ by $3$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (4 x^{4} + 6 x^{2} + 6 x^{2} + 3 \cdot 3)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.1.6

Multiply $3$ by $3$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (4 x^{4} + 6 x^{2} + 6 x^{2} + 9)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.10.2

Add $6 x^{2}$ and $6 x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (4 x^{4} + 12 x^{2} + 9)) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.11

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (x (4 x^{4}) + x (12 x^{2}) + x \cdot 9) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.12

Simplify.

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Step 1.6.12.1

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x \cdot x^{4} + x (12 x^{2}) + x \cdot 9) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.12.2

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x \cdot x^{4} + 12 x \cdot x^{2} + x \cdot 9) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.12.3

Move $9$ to the left of $x$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x \cdot x^{4} + 12 x \cdot x^{2} + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.13

Simplify each term.

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Step 1.6.13.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 1.6.13.1.1

Move $x^{4}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 (x^{4} x) + 12 x \cdot x^{2} + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.13.1.2

Multiply $x^{4}$ by $x$ .

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Step 1.6.13.1.2.1

Raise $x$ to the power of $1$ .

Step 1.6.13.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{4 + 1} + 12 x \cdot x^{2} + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.13.1.3

Add $4$ and $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{5} + 12 x \cdot x^{2} + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.13.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.6.13.2.1

Move $x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{5} + 12 (x^{2} x) + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.13.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.6.13.2.2.1

Raise $x$ to the power of $1$ .

Step 1.6.13.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{5} + 12 x^{2 + 1} + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.13.2.3

Add $2$ and $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{5} + 12 x^{3} + 9 \cdot x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{5} + 12 x^{3} + 9 x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.14

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + B (4 x^{5}) + B (12 x^{3}) + B (9 x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.15

Simplify.

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Step 1.6.15.1

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + B (12 x^{3}) + B (9 x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.15.2

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + B (9 x) + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.15.3

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + \frac{(C x + D) (x^{2} {(2 x^{2} + 3)}^{2})}{{(2 x^{2} + 3)}^{2}} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.16

Cancel the common factor of ${(2 x^{2} + 3)}^{2}$ .

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Step 1.6.16.1

Cancel the common factor.

Step 1.6.16.2

Divide $(C x + D) (x^{2})$ by $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + (C x + D) (x^{2}) + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.17

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x \cdot x^{2} + D x^{2} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.18

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.6.18.1

Move $x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C (x^{2} x) + D x^{2} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.18.2

Multiply $x^{2}$ by $x$ .

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Step 1.6.18.2.1

Raise $x$ to the power of $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C (x^{2} x) + D x^{2} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.18.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{2 + 1} + D x^{2} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.18.3

Add $2$ and $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + \frac{(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})}{2 x^{2} + 3}$

Step 1.6.19

Cancel the common factor of ${(2 x^{2} + 3)}^{2}$ and $2 x^{2} + 3$ .

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Step 1.6.19.1

Factor $2 x^{2} + 3$ out of $(F x + G) (x^{2} {(2 x^{2} + 3)}^{2})$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + \frac{(2 x^{2} + 3) ((F x + G) (x^{2} (2 x^{2} + 3)))}{2 x^{2} + 3}$

Step 1.6.19.2

Cancel the common factors.

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Step 1.6.19.2.1

Multiply by $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + \frac{(2 x^{2} + 3) ((F x + G) (x^{2} (2 x^{2} + 3)))}{(2 x^{2} + 3) \cdot 1}$

Step 1.6.19.2.2

Cancel the common factor.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + \frac{(2 x^{2} + 3) ((F x + G) (x^{2} (2 x^{2} + 3)))}{(2 x^{2} + 3) \cdot 1}$

Step 1.6.19.2.3

Rewrite the expression.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + \frac{(F x + G) (x^{2} (2 x^{2} + 3))}{1}$

Step 1.6.19.2.4

Divide $(F x + G) (x^{2} (2 x^{2} + 3))$ by $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (x^{2} (2 x^{2} + 3))$

Step 1.6.20

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (x^{2} (2 x^{2}) + x^{2} \cdot 3)$

Step 1.6.21

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (2 x^{2} x^{2} + x^{2} \cdot 3)$

Step 1.6.22

Move $3$ to the left of $x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (2 x^{2} x^{2} + 3 \cdot x^{2})$

Step 1.6.23

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.6.23.1

Move $x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (2 (x^{2} x^{2}) + 3 \cdot x^{2})$

Step 1.6.23.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (2 x^{2 + 2} + 3 \cdot x^{2})$

Step 1.6.23.3

Add $2$ and $2$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (2 x^{4} + 3 \cdot x^{2})$

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + (F x + G) (2 x^{4} + 3 x^{2})$

Step 1.6.24

Expand $(F x + G) (2 x^{4} + 3 x^{2})$ using the FOIL Method.

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Step 1.6.24.1

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + F x (2 x^{4} + 3 x^{2}) + G (2 x^{4} + 3 x^{2})$

Step 1.6.24.2

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + F x (2 x^{4}) + F x (3 x^{2}) + G (2 x^{4} + 3 x^{2})$

Step 1.6.24.3

Apply the distributive property.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + F x (2 x^{4}) + F x (3 x^{2}) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25

Simplify each term.

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Step 1.6.25.1

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x \cdot x^{4} + F x (3 x^{2}) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 1.6.25.2.1

Move $x^{4}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F (x^{4} x) + F x (3 x^{2}) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.2.2

Multiply $x^{4}$ by $x$ .

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Step 1.6.25.2.2.1

Raise $x$ to the power of $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F (x^{4} x) + F x (3 x^{2}) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{4 + 1} + F x (3 x^{2}) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.2.3

Add $4$ and $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + F x (3 x^{2}) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.3

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x \cdot x^{2} + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.6.25.4.1

Move $x^{2}$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F (x^{2} x) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.4.2

Multiply $x^{2}$ by $x$ .

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Step 1.6.25.4.2.1

Raise $x$ to the power of $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F (x^{2} x) + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{2 + 1} + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.4.3

Add $2$ and $1$ .

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + G (2 x^{4}) + G (3 x^{2})$

Step 1.6.25.5

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + G (3 x^{2})$

Step 1.6.25.6

Rewrite using the commutative property of multiplication.

$x^{4} + x^{2} + 1 = 4 A x^{4} + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7

Reorder.

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Step 1.7.1

Move $A$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 A x^{2} + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7.2

Move $A$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 B x^{5} + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7.3

Move $B$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 B x^{3} + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7.4

Move $B$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 x^{3} B + 9 B x + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7.5

Move $B$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 x^{3} B + 9 x B + C x^{3} + D x^{2} + 2 F x^{5} + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7.6

Move $F$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 x^{3} B + 9 x B + C x^{3} + D x^{2} + 2 x^{5} F + 3 F x^{3} + 2 G x^{4} + 3 G x^{2}$

Step 1.7.7

Move $F$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 x^{3} B + 9 x B + C x^{3} + D x^{2} + 2 x^{5} F + 3 x^{3} F + 2 G x^{4} + 3 G x^{2}$

Step 1.7.8

Move $G$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 x^{3} B + 9 x B + C x^{3} + D x^{2} + 2 x^{5} F + 3 x^{3} F + 2 x^{4} G + 3 G x^{2}$

Step 1.7.9

Move $G$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 9 A + 4 x^{5} B + 12 x^{3} B + 9 x B + C x^{3} + D x^{2} + 2 x^{5} F + 3 x^{3} F + 2 x^{4} G + 3 x^{2} G$

Step 1.7.10

Move $9 A$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 4 x^{5} B + 12 x^{3} B + 9 x B + C x^{3} + D x^{2} + 2 x^{5} F + 3 x^{3} F + 2 x^{4} G + 3 x^{2} G + 9 A$

Step 1.7.11

Move $9 x B$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 4 x^{5} B + 12 x^{3} B + C x^{3} + D x^{2} + 2 x^{5} F + 3 x^{3} F + 2 x^{4} G + 3 x^{2} G + 9 x B + 9 A$

Step 1.7.12

Move $D x^{2}$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 12 x^{2} A + 4 x^{5} B + 12 x^{3} B + C x^{3} + 2 x^{5} F + 3 x^{3} F + 2 x^{4} G + D x^{2} + 3 x^{2} G + 9 x B + 9 A$

Step 1.7.13

Move $12 x^{2} A$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 4 x^{5} B + 12 x^{3} B + C x^{3} + 2 x^{5} F + 3 x^{3} F + 2 x^{4} G + 12 x^{2} A + D x^{2} + 3 x^{2} G + 9 x B + 9 A$

Step 1.7.14

Move $3 x^{3} F$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 4 x^{5} B + 12 x^{3} B + C x^{3} + 2 x^{5} F + 2 x^{4} G + 3 x^{3} F + 12 x^{2} A + D x^{2} + 3 x^{2} G + 9 x B + 9 A$

Step 1.7.15

Move $C x^{3}$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 4 x^{5} B + 12 x^{3} B + 2 x^{5} F + 2 x^{4} G + C x^{3} + 3 x^{3} F + 12 x^{2} A + D x^{2} + 3 x^{2} G + 9 x B + 9 A$

Step 1.7.16

Move $12 x^{3} B$ .

$x^{4} + x^{2} + 1 = 4 x^{4} A + 4 x^{5} B + 2 x^{5} F + 2 x^{4} G + 12 x^{3} B + C x^{3} + 3 x^{3} F + 12 x^{2} A + D x^{2} + 3 x^{2} G + 9 x B + 9 A$

Step 1.7.17

Move $4 x^{4} A$ .

$x^{4} + x^{2} + 1 = 4 x^{5} B + 2 x^{5} F + 4 x^{4} A + 2 x^{4} G + 12 x^{3} B + C x^{3} + 3 x^{3} F + 12 x^{2} A + D x^{2} + 3 x^{2} G + 9 x B + 9 A$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{5}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 4 B + 2 F$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{4}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 4 A + 2 G$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 12 B + C + 3 F$

Step 2.4

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 12 A + D + 3 G$

Step 2.5

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 9 B$

Step 2.6

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 9 A$

Step 2.7

Set up the system of equations to find the coefficients of the partial fractions.

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$0 = 9 B$

$1 = 9 A$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$0 = 9 B$

$1 = 9 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $B$ in $0 = 9 B$ .

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Step 3.1.1

Rewrite the equation as $9 B = 0$ .

$9 B = 0$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.1.2

Divide each term in $9 B = 0$ by $9$ and simplify.

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Step 3.1.2.1

Divide each term in $9 B = 0$ by $9$ .

$\frac{9 B}{9} = \frac{0}{9}$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $9$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{9 B}{9} = \frac{0}{9}$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.1.2.2.1.2

Divide $B$ by $1$ .

$B = \frac{0}{9}$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$B = \frac{0}{9}$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$B = \frac{0}{9}$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $0$ by $9$ .

$B = 0$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$B = 0$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$B = 0$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$B = 0$

$0 = 4 B + 2 F$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2

Replace all occurrences of $B$ with $0$ in each equation.

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Step 3.2.1

Replace all occurrences of $B$ in $0 = 4 B + 2 F$ with $0$ .

$0 = 4 (0) + 2 F$

$B = 0$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $4 (0) + 2 F$ .

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Step 3.2.2.1.1

Multiply $4$ by $0$ .

$0 = 0 + 2 F$

$B = 0$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2.2.1.2

Add $0$ and $2 F$ .

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$0 = 12 B + C + 3 F$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2.3

Replace all occurrences of $B$ in $0 = 12 B + C + 3 F$ with $0$ .

$0 = 12 (0) + C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $12 (0) + C + 3 F$ .

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Step 3.2.4.1.1

Multiply $12$ by $0$ .

$0 = 0 + C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2.4.1.2

Add $0$ and $C$ .

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$1 = 9 A$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$1 = 9 A$

Step 3.2.5

Rewrite the equation as $9 A = 1$ .

$9 A = 1$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.2.6

Divide each term in $9 A = 1$ by $9$ and simplify.

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Step 3.2.6.1

Divide each term in $9 A = 1$ by $9$ .

$\frac{9 A}{9} = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.2.6.2

Simplify the left side.

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Step 3.2.6.2.1

Cancel the common factor of $9$ .

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Step 3.2.6.2.1.1

Cancel the common factor.

$\frac{9 A}{9} = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.2.6.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$A = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$A = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$A = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$A = \frac{1}{9}$

$0 = C + 3 F$

$0 = 2 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.3

Replace all occurrences of $A$ with $\frac{1}{9}$ in each equation.

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Step 3.3.1

Rewrite the equation as $2 F = 0$ .

$2 F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.3.2

Divide each term in $2 F = 0$ by $2$ and simplify.

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Step 3.3.2.1

Divide each term in $2 F = 0$ by $2$ .

$\frac{2 F}{2} = \frac{0}{2}$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{2 F}{2} = \frac{0}{2}$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.3.2.2.1.2

Divide $F$ by $1$ .

$F = \frac{0}{2}$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$F = \frac{0}{2}$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$F = \frac{0}{2}$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $0$ by $2$ .

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 A + 2 G$

$1 = 12 A + D + 3 G$

Step 3.3.3

Replace all occurrences of $A$ in $1 = 4 A + 2 G$ with $\frac{1}{9}$ .

$1 = 4 (\frac{1}{9}) + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 12 A + D + 3 G$

Step 3.3.4

Simplify the right side.

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Step 3.3.4.1

Combine $4$ and $\frac{1}{9}$ .

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 12 A + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 12 A + D + 3 G$

Step 3.3.5

Replace all occurrences of $A$ in $1 = 12 A + D + 3 G$ with $\frac{1}{9}$ .

$1 = 12 (\frac{1}{9}) + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.3.6

Simplify the right side.

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Step 3.3.6.1

Simplify each term.

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Step 3.3.6.1.1

Cancel the common factor of $3$ .

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Step 3.3.6.1.1.1

Factor $3$ out of $12$ .

$1 = 3 (4) (\frac{1}{9}) + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.3.6.1.1.2

Factor $3$ out of $9$ .

$1 = 3 \cdot (4 (\frac{1}{3 \cdot 3})) + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.3.6.1.1.3

Cancel the common factor.

$1 = 3 \cdot (4 (\frac{1}{3 \cdot 3})) + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.3.6.1.1.4

Rewrite the expression.

$1 = 4 (\frac{1}{3}) + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = 4 (\frac{1}{3}) + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.3.6.1.2

Combine $4$ and $\frac{1}{3}$ .

$1 = \frac{4}{3} + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = \frac{4}{3} + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = \frac{4}{3} + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$1 = \frac{4}{3} + D + 3 G$

$1 = \frac{4}{9} + 2 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4

Replace all occurrences of $F$ with $0$ in each equation.

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Step 3.4.1

Rewrite the equation as $\frac{4}{9} + 2 G = 1$ .

$\frac{4}{9} + 2 G = 1$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.2

Move all terms not containing $G$ to the right side of the equation.

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Step 3.4.2.1

Subtract $\frac{4}{9}$ from both sides of the equation.

$2 G = 1 - \frac{4}{9}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.2.2

Write $1$ as a fraction with a common denominator.

$2 G = \frac{9}{9} - \frac{4}{9}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.2.3

Combine the numerators over the common denominator.

$2 G = \frac{9 - 4}{9}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.2.4

Subtract $4$ from $9$ .

$2 G = \frac{5}{9}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$2 G = \frac{5}{9}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.3

Divide each term in $2 G = \frac{5}{9}$ by $2$ and simplify.

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Step 3.4.3.1

Divide each term in $2 G = \frac{5}{9}$ by $2$ .

$\frac{2 G}{2} = \frac{\frac{5}{9}}{2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $2$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

$\frac{2 G}{2} = \frac{\frac{5}{9}}{2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.3.2.1.2

Divide $G$ by $1$ .

$G = \frac{\frac{5}{9}}{2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$G = \frac{\frac{5}{9}}{2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$G = \frac{\frac{5}{9}}{2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$G = \frac{5}{9} \cdot \frac{1}{2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.3.3.2

Multiply $\frac{5}{9} \cdot \frac{1}{2}$ .

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Step 3.4.3.3.2.1

Multiply $\frac{5}{9}$ by $\frac{1}{2}$ .

$G = \frac{5}{9 \cdot 2}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.3.3.2.2

Multiply $9$ by $2$ .

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$0 = C + 3 F$

$B = 0$

Step 3.4.4

Replace all occurrences of $F$ in $0 = C + 3 F$ with $0$ .

$0 = C + 3 (0)$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.4.5

Simplify the right side.

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Step 3.4.5.1

Simplify $C + 3 (0)$ .

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Step 3.4.5.1.1

Multiply $3$ by $0$ .

$0 = C + 0$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.4.5.1.2

Add $C$ and $0$ .

$0 = C$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$0 = C$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$0 = C$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$0 = C$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5

Replace all occurrences of $G$ with $\frac{5}{18}$ in each equation.

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Step 3.5.1

Rewrite the equation as $C = 0$ .

$C = 0$

$G = \frac{5}{18}$

$1 = \frac{4}{3} + D + 3 G$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.2

Replace all occurrences of $G$ in $1 = \frac{4}{3} + D + 3 G$ with $\frac{5}{18}$ .

$1 = \frac{4}{3} + D + 3 (\frac{5}{18})$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify $\frac{4}{3} + D + 3 (\frac{5}{18})$ .

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Step 3.5.3.1.1

Cancel the common factor of $3$ .

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Step 3.5.3.1.1.1

Factor $3$ out of $18$ .

$1 = \frac{4}{3} + D + 3 (\frac{5}{3 (6)})$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.1.2

Cancel the common factor.

$1 = \frac{4}{3} + D + 3 (\frac{5}{3 \cdot 6})$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.1.3

Rewrite the expression.

$1 = \frac{4}{3} + D + \frac{5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$1 = \frac{4}{3} + D + \frac{5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.2

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1 = D + \frac{4}{3} \cdot \frac{2}{2} + \frac{5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.3.1.3.1

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$1 = D + \frac{4 \cdot 2}{3 \cdot 2} + \frac{5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.3.2

Multiply $3$ by $2$ .

$1 = D + \frac{4 \cdot 2}{6} + \frac{5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$1 = D + \frac{4 \cdot 2}{6} + \frac{5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.4

Combine the numerators over the common denominator.

$1 = D + \frac{4 \cdot 2 + 5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.5

Simplify the numerator.

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Step 3.5.3.1.5.1

Multiply $4$ by $2$ .

$1 = D + \frac{8 + 5}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.5.3.1.5.2

Add $8$ and $5$ .

$1 = D + \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$1 = D + \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$1 = D + \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$1 = D + \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$1 = D + \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.6

Solve for $D$ in $1 = D + \frac{13}{6}$ .

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Step 3.6.1

Rewrite the equation as $D + \frac{13}{6} = 1$ .

$D + \frac{13}{6} = 1$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.6.2

Move all terms not containing $D$ to the right side of the equation.

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Step 3.6.2.1

Subtract $\frac{13}{6}$ from both sides of the equation.

$D = 1 - \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.6.2.2

Write $1$ as a fraction with a common denominator.

$D = \frac{6}{6} - \frac{13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.6.2.3

Combine the numerators over the common denominator.

$D = \frac{6 - 13}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.6.2.4

Subtract $13$ from $6$ .

$D = \frac{- 7}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.6.2.5

Move the negative in front of the fraction.

$D = - \frac{7}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$D = - \frac{7}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

$D = - \frac{7}{6}$

$C = 0$

$G = \frac{5}{18}$

$F = 0$

$A = \frac{1}{9}$

$B = 0$

Step 3.7

Solve the system of equations.

$D = - \frac{7}{6} C = 0 G = \frac{5}{18} F = 0 A = \frac{1}{9} B = 0$

Step 3.8

List all of the solutions.

$D = - \frac{7}{6}, C = 0, G = \frac{5}{18}, F = 0, A = \frac{1}{9}, B = 0$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x^{2}} + \frac{B}{x} + \frac{C x + D}{{(2 x^{2} + 3)}^{2}} + \frac{F x + G}{2 x^{2} + 3}$ with the values found for $A$ , $B$ , $C$ , $D$ , $F$ , and $G$ .

$\frac{\frac{1}{9}}{x^{2}} + \frac{0}{x} + \frac{0 x - \frac{7}{6}}{{(2 x^{2} + 3)}^{2}} + \frac{0 x + \frac{5}{18}}{2 x^{2} + 3}$

Step 5

Simplify.

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Step 5.1

Multiply $0$ by $x$ .

$\frac{\frac{1}{9}}{x^{2}} + \frac{0}{x} + \frac{0 - \frac{7}{6}}{{(2 x^{2} + 3)}^{2}} + \frac{(0) \cdot x + \frac{5}{18}}{2 x^{2} + 3}$

Step 5.2

Subtract $\frac{7}{6}$ from $0$ .

$\frac{\frac{1}{9}}{x^{2}} + \frac{0}{x} + \frac{- \frac{7}{6}}{{(2 x^{2} + 3)}^{2}} + \frac{(0) \cdot x + \frac{5}{18}}{2 x^{2} + 3}$

Step 5.3

Multiply $0$ by $x$ .

$\frac{\frac{1}{9}}{x^{2}} + \frac{0}{x} + \frac{- \frac{7}{6}}{{(2 x^{2} + 3)}^{2}} + \frac{0 + \frac{5}{18}}{2 x^{2} + 3}$

Step 5.4

Add $0$ and $\frac{5}{18}$ .

$\frac{\frac{1}{9}}{x^{2}} + \frac{0}{x} + \frac{- \frac{7}{6}}{{(2 x^{2} + 3)}^{2}} + \frac{\frac{5}{18}}{2 x^{2} + 3}$