Simplify ((x^4+x^3-30x^2)/(x^2-3x-18))÷((x^3+x^2-30x)/(x^2-36))

$\frac{x^{4} + x^{3} - 30 x^{2}}{x^{2} - 3 x - 18} \div \frac{x^{3} + x^{2} - 30 x}{x^{2} - 36}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{4} + x^{3} - 30 x^{2}}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

Simplify the numerator.

Factor $x^{2}$ out of $x^{4} + x^{3} - 30 x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$\frac{x^{2} x^{2} + x^{3} - 30 x^{2}}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

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

$\frac{x^{2} x^{2} + x^{2} x - 30 x^{2}}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

Factor $x^{2}$ out of $- 30 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} x + x^{2} \cdot - 30}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

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

$\frac{x^{2} (x^{2} + x) + x^{2} \cdot - 30}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

Factor $x^{2}$ out of $x^{2} (x^{2} + x) + x^{2} \cdot - 30$ .

$\frac{x^{2} (x^{2} + x - 30)}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

Factor $x^{2} + x - 30$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$\frac{x^{2} ((x - 5) (x + 6))}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

$\frac{x^{2} (x - 5) (x + 6)}{x^{2} - 3 x - 18} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

Factor $x^{2} - 3 x - 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Write the factored form using these integers.

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{x^{2} - 36}{x^{3} + x^{2} - 30 x}$

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{x^{2} - 6^{2}}{x^{3} + x^{2} - 30 x}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{(x + 6) (x - 6)}{x^{3} + x^{2} - 30 x}$

Simplify the denominator.

Factor $x$ out of $x^{3} + x^{2} - 30 x$ .

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

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{(x + 6) (x - 6)}{x \cdot x^{2} + x^{2} - 30 x}$

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

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{(x + 6) (x - 6)}{x \cdot x^{2} + x \cdot x - 30 x}$

Factor $x$ out of $- 30 x$ .

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{(x + 6) (x - 6)}{x \cdot x^{2} + x \cdot x + x \cdot - 30}$

Factor $x$ out of $x \cdot x^{2} + x \cdot x$ .

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{(x + 6) (x - 6)}{x (x^{2} + x) + x \cdot - 30}$

Factor $x$ out of $x (x^{2} + x) + x \cdot - 30$ .

$\frac{x^{2} (x - 5) (x + 6)}{(x - 6) (x + 3)} \cdot \frac{(x + 6) (x - 6)}{x (x^{2} + x - 30)}$

Factor $x^{2} + x - 30$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

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

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

Simplify terms.

Combine.

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

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

Factor $x$ out of $x^{2} (x - 5) (x + 6) ((x + 6) (x - 6))$ .

$\frac{x (((x (x - 5)) (x + 6)) ((x + 6) (x - 6)))}{(x - 6) (x + 3) (x (x - 5) (x + 6))}$

Cancel the common factors.

Factor $x$ out of $(x - 6) (x + 3) (x (x - 5) (x + 6))$ .

$\frac{x (((x (x - 5)) (x + 6)) ((x + 6) (x - 6)))}{x ((x - 6) (x + 3) ((x - 5) (x + 6)))}$

Cancel the common factor.

$\frac{x (((x (x - 5)) (x + 6)) ((x + 6) (x - 6)))}{x ((x - 6) (x + 3) ((x - 5) (x + 6)))}$

Rewrite the expression.

$\frac{((x (x - 5)) (x + 6)) ((x + 6) (x - 6))}{(x - 6) (x + 3) ((x - 5) (x + 6))}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{x (x - 5) (x + 6) ((x + 6) (x - 6))}{(x - 6) (x + 3) ((x - 5) (x + 6))}$

Rewrite the expression.

$\frac{((x) (x + 6)) ((x + 6) (x - 6))}{(x - 6) (x + 3) (x + 6)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{x (x + 6) ((x + 6) (x - 6))}{(x - 6) (x + 3) (x + 6)}$

Rewrite the expression.

$\frac{(x) ((x + 6) (x - 6))}{(x - 6) (x + 3)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{x ((x + 6) (x - 6))}{(x - 6) (x + 3)}$

Rewrite the expression.

$\frac{(x) (x + 6)}{x + 3}$

$\frac{x (x + 6)}{x + 3}$

Simplify ((x^4+x^3-30x^2)/(x^2-3x-18))÷((x^3+x^2-30x)/(x^2-36))

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