Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $3 (x + 1) x^{2}$ .

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Step 1.1

Multiply $\frac{\frac{4}{3 x} + \frac{2}{x^{2}}}{\frac{x}{x + 1} - \frac{4}{x + 1}}$ by $\frac{3 (x + 1) x^{2}}{3 (x + 1) x^{2}}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3 x$ .

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Step 3.1.1

Factor $3 x$ out of $3 (x + 1) x^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

Factor $x^{2}$ out of $3 (x + 1) x^{2}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Multiply $2$ by $3$ .

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

Step 3.4

Cancel the common factor of $x + 1$ .

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Step 3.4.1

Factor $x + 1$ out of $3 (x + 1) x^{2}$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Add $1$ and $2$ .

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

Step 3.8

Cancel the common factor of $x + 1$ .

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Step 3.8.1

Move the leading negative in $- \frac{4}{x + 1}$ into the numerator.

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

Step 3.8.2

Factor $x + 1$ out of $3 (x + 1) x^{2}$ .

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

Step 3.8.3

Cancel the common factor.

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

Step 3.8.4

Rewrite the expression.

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

Step 3.9

Multiply $- 4$ by $3$ .

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $(x + 1) \cdot 2$ out of $(x + 1) x \cdot 4 + 6 (x + 1)$ .

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Step 4.1.1

Factor $(x + 1) \cdot 2$ out of $(x + 1) x \cdot 4$ .

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

Step 4.1.2

Factor $(x + 1) \cdot 2$ out of $6 (x + 1)$ .

$\frac{(x + 1) \cdot 2 (x \cdot 2) + (x + 1) \cdot 2 (3)}{3 x^{3} - 12 x^{2}}$

Step 4.1.3

Factor $(x + 1) \cdot 2$ out of $(x + 1) \cdot 2 (x \cdot 2) + (x + 1) \cdot 2 (3)$ .

$\frac{(x + 1) \cdot 2 (x \cdot 2 + 3)}{3 x^{3} - 12 x^{2}}$

Step 4.2

Move $2$ to the left of $x$ .

$\frac{(x + 1) \cdot 2 (2 x + 3)}{3 x^{3} - 12 x^{2}}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $3 x^{2}$ out of $3 x^{3} - 12 x^{2}$ .

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Step 5.1.1

Factor $3 x^{2}$ out of $3 x^{3}$ .

$\frac{(x + 1) \cdot 2 (2 x + 3)}{3 x^{2} (x) - 12 x^{2}}$

Step 5.1.2

Factor $3 x^{2}$ out of $- 12 x^{2}$ .

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

Step 5.1.3

Factor $3 x^{2}$ out of $3 x^{2} (x) + 3 x^{2} (- 4)$ .

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

Step 5.2

Move $2$ to the left of $x + 1$ .

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