Calculus Examples

$y = \frac{(x - 1) (x - 4)}{(x - 2) (x - 3)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = (x - 1) (x - 4)$ and $g (x) = (x - 2) (x - 3)$ .

$\frac{(x - 2) (x - 3) \frac{d}{d x} [(x - 1) (x - 4)] - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x - 4$ .

$\frac{(x - 2) (x - 3) ((x - 1) \frac{d}{d x} [x - 4] + (x - 4) \frac{d}{d x} [x - 1]) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{(x - 2) (x - 3) ((x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x - 1]) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(x - 2) (x - 3) ((x - 1) (1 + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x - 1]) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{(x - 2) (x - 3) ((x - 1) (1 + 0) + (x - 4) \frac{d}{d x} [x - 1]) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$\frac{(x - 2) (x - 3) ((x - 1) \cdot 1 + (x - 4) \frac{d}{d x} [x - 1]) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.4.2

Multiply $x - 1$ by $1$ .

$\frac{(x - 2) (x - 3) (x - 1 + (x - 4) \frac{d}{d x} [x - 1]) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.5

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{(x - 2) (x - 3) (x - 1 + (x - 4) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(x - 2) (x - 3) (x - 1 + (x - 4) (1 + \frac{d}{d x} [- 1])) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{(x - 2) (x - 3) (x - 1 + (x - 4) (1 + 0)) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.8

Simplify by adding terms.

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Step 3.8.1

Add $1$ and $0$ .

$\frac{(x - 2) (x - 3) (x - 1 + (x - 4) \cdot 1) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.8.2

Multiply $x - 4$ by $1$ .

$\frac{(x - 2) (x - 3) (x - 1 + x - 4) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.8.3

Add $x$ and $x$ .

$\frac{(x - 2) (x - 3) (2 x - 1 - 4) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 3.8.4

Subtract $4$ from $- 1$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) \frac{d}{d x} [(x - 2) (x - 3)]}{{((x - 2) (x - 3))}^{2}}$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 2$ and $g (x) = x - 3$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) ((x - 2) \frac{d}{d x} [x - 3] + (x - 3) \frac{d}{d x} [x - 2])}{{((x - 2) (x - 3))}^{2}}$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) ((x - 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [x - 2])}{{((x - 2) (x - 3))}^{2}}$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) ((x - 2) (1 + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [x - 2])}{{((x - 2) (x - 3))}^{2}}$

Step 5.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) ((x - 2) (1 + 0) + (x - 3) \frac{d}{d x} [x - 2])}{{((x - 2) (x - 3))}^{2}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $1$ and $0$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) ((x - 2) \cdot 1 + (x - 3) \frac{d}{d x} [x - 2])}{{((x - 2) (x - 3))}^{2}}$

Step 5.4.2

Multiply $x - 2$ by $1$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (x - 2 + (x - 3) \frac{d}{d x} [x - 2])}{{((x - 2) (x - 3))}^{2}}$

Step 5.5

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (x - 2 + (x - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]))}{{((x - 2) (x - 3))}^{2}}$

Step 5.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (x - 2 + (x - 3) (1 + \frac{d}{d x} [- 2]))}{{((x - 2) (x - 3))}^{2}}$

Step 5.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (x - 2 + (x - 3) (1 + 0))}{{((x - 2) (x - 3))}^{2}}$

Step 5.8

Simplify by adding terms.

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Step 5.8.1

Add $1$ and $0$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (x - 2 + (x - 3) \cdot 1)}{{((x - 2) (x - 3))}^{2}}$

Step 5.8.2

Multiply $x - 3$ by $1$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (x - 2 + x - 3)}{{((x - 2) (x - 3))}^{2}}$

Step 5.8.3

Add $x$ and $x$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (2 x - 2 - 3)}{{((x - 2) (x - 3))}^{2}}$

Step 5.8.4

Subtract $3$ from $- 2$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (2 x - 5)}{{((x - 2) (x - 3))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the product rule to $(x - 2) (x - 3)$ .

$\frac{(x - 2) (x - 3) (2 x - 5) - (x - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{(x - 2) (x - 3) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Expand $(x - 2) (x - 3)$ using the FOIL Method.

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Step 6.3.1.1.1

Apply the distributive property.

$\frac{(x (x - 3) - 2 (x - 3)) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.1.2

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 3 - 2 (x - 3)) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.1.3

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 3 - 2 x - 2 \cdot - 3) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.2

Simplify and combine like terms.

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Step 6.3.1.2.1

Simplify each term.

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Step 6.3.1.2.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot - 3 - 2 x - 2 \cdot - 3) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.2.1.2

Move $- 3$ to the left of $x$ .

$\frac{(x^{2} - 3 \cdot x - 2 x - 2 \cdot - 3) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.2.1.3

Multiply $- 2$ by $- 3$ .

$\frac{(x^{2} - 3 x - 2 x + 6) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.2.2

Subtract $2 x$ from $- 3 x$ .

$\frac{(x^{2} - 5 x + 6) (2 x - 5) + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.3

Expand $(x^{2} - 5 x + 6) (2 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (2 x) + x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4

Simplify each term.

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Step 6.3.1.4.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.2

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

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Step 6.3.1.4.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.2.2

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

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Step 6.3.1.4.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot - 5 - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.3

Move $- 5$ to the left of $x^{2}$ .

$\frac{2 x^{3} - 5 \cdot x^{2} - 5 x (2 x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.4

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 5 x^{2} - 5 \cdot 2 x \cdot x - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.4.5.1

Move $x$ .

$\frac{2 x^{3} - 5 x^{2} - 5 \cdot 2 (x \cdot x) - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.5.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 5 x^{2} - 5 \cdot 2 x^{2} - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.6

Multiply $- 5$ by $2$ .

$\frac{2 x^{3} - 5 x^{2} - 10 x^{2} - 5 x \cdot - 5 + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.7

Multiply $- 5$ by $- 5$ .

$\frac{2 x^{3} - 5 x^{2} - 10 x^{2} + 25 x + 6 (2 x) + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.8

Multiply $2$ by $6$ .

$\frac{2 x^{3} - 5 x^{2} - 10 x^{2} + 25 x + 12 x + 6 \cdot - 5 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.4.9

Multiply $6$ by $- 5$ .

$\frac{2 x^{3} - 5 x^{2} - 10 x^{2} + 25 x + 12 x - 30 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.5

Subtract $10 x^{2}$ from $- 5 x^{2}$ .

$\frac{2 x^{3} - 15 x^{2} + 25 x + 12 x - 30 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.6

Add $25 x$ and $12 x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x - - 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.7

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x + 1) (x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.8

Expand $(- x + 1) (x - 4)$ using the FOIL Method.

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Step 6.3.1.8.1

Apply the distributive property.

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x (x - 4) + 1 (x - 4)) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.8.2

Apply the distributive property.

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x \cdot x - x \cdot - 4 + 1 (x - 4)) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.8.3

Apply the distributive property.

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x \cdot x - x \cdot - 4 + 1 x + 1 \cdot - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.9

Simplify and combine like terms.

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Step 6.3.1.9.1

Simplify each term.

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Step 6.3.1.9.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.9.1.1.1

Move $x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- (x \cdot x) - x \cdot - 4 + 1 x + 1 \cdot - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.9.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x^{2} - x \cdot - 4 + 1 x + 1 \cdot - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.9.1.2

Multiply $- 4$ by $- 1$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x^{2} + 4 x + 1 x + 1 \cdot - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.9.1.3

Multiply $x$ by $1$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x^{2} + 4 x + x + 1 \cdot - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.9.1.4

Multiply $- 4$ by $1$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x^{2} + 4 x + x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.9.2

Add $4 x$ and $x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 + (- x^{2} + 5 x - 4) (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.10

Expand $(- x^{2} + 5 x - 4) (2 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - x^{2} (2 x) - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11

Simplify each term.

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Step 6.3.1.11.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 1 \cdot 2 x^{2} x - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.2

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

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Step 6.3.1.11.2.1

Move $x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.2.2

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

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Step 6.3.1.11.2.2.1

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

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.2.2.2

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

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 1 \cdot 2 x^{3} - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} - x^{2} \cdot - 5 + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.4

Multiply $- 5$ by $- 1$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 5 x (2 x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 5 \cdot 2 x \cdot x + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.11.6.1

Move $x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 5 \cdot 2 (x \cdot x) + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 5 \cdot 2 x^{2} + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.7

Multiply $5$ by $2$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 10 x^{2} + 5 x \cdot - 5 - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.8

Multiply $- 5$ by $5$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 10 x^{2} - 25 x - 4 (2 x) - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.9

Multiply $2$ by $- 4$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 10 x^{2} - 25 x - 8 x - 4 \cdot - 5}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.11.10

Multiply $- 4$ by $- 5$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 5 x^{2} + 10 x^{2} - 25 x - 8 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.12

Add $5 x^{2}$ and $10 x^{2}$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 15 x^{2} - 25 x - 8 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.1.13

Subtract $8 x$ from $- 25 x$ .

$\frac{2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 15 x^{2} - 33 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.2

Combine the opposite terms in $2 x^{3} - 15 x^{2} + 37 x - 30 - 2 x^{3} + 15 x^{2} - 33 x + 20$ .

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Step 6.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{- 15 x^{2} + 37 x - 30 + 0 + 15 x^{2} - 33 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.2.2

Add $- 15 x^{2} + 37 x - 30$ and $0$ .

$\frac{- 15 x^{2} + 37 x - 30 + 15 x^{2} - 33 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.2.3

Add $- 15 x^{2}$ and $15 x^{2}$ .

$\frac{37 x - 30 + 0 - 33 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.2.4

Add $37 x - 30$ and $0$ .

$\frac{37 x - 30 - 33 x + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.3

Subtract $33 x$ from $37 x$ .

$\frac{4 x - 30 + 20}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.3.4

Add $- 30$ and $20$ .

$\frac{4 x - 10}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.4

Factor $2$ out of $4 x - 10$ .

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Step 6.4.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) - 10}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.4.2

Factor $2$ out of $- 10$ .

$\frac{2 (2 x) + 2 (- 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$

Step 6.4.3

Factor $2$ out of $2 (2 x) + 2 (- 5)$ .

$\frac{2 (2 x - 5)}{{(x - 2)}^{2} {(x - 3)}^{2}}$