Simplify ((36-x^2)/(6-x)*(x^2-2x-48)/(x^3-2x^2-48x))÷((2x^3+14x^2+12x)/(x^3-9x^2-10x))

$\frac{36 - x^{2}}{6 - x} \cdot \frac{x^{2} - 2 x - 48}{x^{3} - 2 x^{2} - 48 x} \div \frac{2 x^{3} + 14 x^{2} + 12 x}{x^{3} - 9 x^{2} - 10 x}$

To divide by a fraction, multiply by its reciprocal.

$\frac{36 - x^{2}}{6 - x} \cdot \frac{x^{2} - 2 x - 48}{x^{3} - 2 x^{2} - 48 x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Simplify the numerator.

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

$\frac{6^{2} - x^{2}}{6 - x} \cdot \frac{x^{2} - 2 x - 48}{x^{3} - 2 x^{2} - 48 x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

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

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{x^{2} - 2 x - 48}{x^{3} - 2 x^{2} - 48 x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x^{2} - 2 x - 48$ 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 $- 48$ and whose sum is $- 2$ .

$- 8, 6$

Write the factored form using these integers.

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x^{3} - 2 x^{2} - 48 x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Simplify the denominator.

Factor $x$ out of $x^{3} - 2 x^{2} - 48 x$ .

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

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x \cdot x^{2} - 2 x^{2} - 48 x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

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

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x \cdot x^{2} + x (- 2 x) - 48 x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x$ out of $- 48 x$ .

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x \cdot x^{2} + x (- 2 x) + x \cdot - 48} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

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

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x (x^{2} - 2 x) + x \cdot - 48} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x$ out of $x (x^{2} - 2 x) + x \cdot - 48$ .

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x (x^{2} - 2 x - 48)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x^{2} - 2 x - 48$ 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 $- 48$ and whose sum is $- 2$ .

$- 8, 6$

Write the factored form using these integers.

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x ((x - 8) (x + 6))} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x (x - 8) (x + 6)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Simplify terms.

Cancel the common factor of $6 - x$ .

Cancel the common factor.

$\frac{(6 + x) (6 - x)}{6 - x} \cdot \frac{(x - 8) (x + 6)}{x (x - 8) (x + 6)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Divide $6 + x$ by $1$ .

$(6 + x) \cdot \frac{(x - 8) (x + 6)}{x (x - 8) (x + 6)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$(6 + x) \cdot \frac{(x - 8) (x + 6)}{x (x - 8) (x + 6)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Rewrite the expression.

$(6 + x) \cdot \frac{x + 6}{x (x + 6)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$(6 + x) \cdot \frac{x + 6}{x (x + 6)} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Rewrite the expression.

$(6 + x) \cdot \frac{1}{x} \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Multiply $6 + x$ by $\frac{1}{x}$ .

$\frac{6 + x}{x} \cdot \frac{x^{3} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Simplify the numerator.

Factor $x$ out of $x^{3} - 9 x^{2} - 10 x$ .

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

$\frac{6 + x}{x} \cdot \frac{x \cdot x^{2} - 9 x^{2} - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x$ out of $- 9 x^{2}$ .

$\frac{6 + x}{x} \cdot \frac{x \cdot x^{2} + x (- 9 x) - 10 x}{2 x^{3} + 14 x^{2} + 12 x}$

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

$\frac{6 + x}{x} \cdot \frac{x \cdot x^{2} + x (- 9 x) + x \cdot - 10}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x$ out of $x \cdot x^{2} + x (- 9 x)$ .

$\frac{6 + x}{x} \cdot \frac{x (x^{2} - 9 x) + x \cdot - 10}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x$ out of $x (x^{2} - 9 x) + x \cdot - 10$ .

$\frac{6 + x}{x} \cdot \frac{x (x^{2} - 9 x - 10)}{2 x^{3} + 14 x^{2} + 12 x}$

Factor $x^{2} - 9 x - 10$ 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 $- 10$ and whose sum is $- 9$ .

$- 10, 1$

Write the factored form using these integers.

$\frac{6 + x}{x} \cdot \frac{x ((x - 10) (x + 1))}{2 x^{3} + 14 x^{2} + 12 x}$

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x^{3} + 14 x^{2} + 12 x}$

Simplify the denominator.

Factor $2 x$ out of $2 x^{3} + 14 x^{2} + 12 x$ .

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

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x (x^{2}) + 14 x^{2} + 12 x}$

Factor $2 x$ out of $14 x^{2}$ .

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x (x^{2}) + 2 x (7 x) + 12 x}$

Factor $2 x$ out of $12 x$ .

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x (x^{2}) + 2 x (7 x) + 2 x (6)}$

Factor $2 x$ out of $2 x (x^{2}) + 2 x (7 x)$ .

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x (x^{2} + 7 x) + 2 x (6)}$

Factor $2 x$ out of $2 x (x^{2} + 7 x) + 2 x (6)$ .

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x (x^{2} + 7 x + 6)}$

Factor $x^{2} + 7 x + 6$ 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 $6$ and whose sum is $7$ .

$1, 6$

Write the factored form using these integers.

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x ((x + 1) (x + 6))}$

$\frac{6 + x}{x} \cdot \frac{x (x - 10) (x + 1)}{2 x (x + 1) (x + 6)}$

Simplify terms.

Cancel the common factor of $x$ .

Factor $x$ out of $x (x - 10) (x + 1)$ .

$\frac{6 + x}{x} \cdot \frac{x ((x - 10) (x + 1))}{2 x (x + 1) (x + 6)}$

Cancel the common factor.

$\frac{6 + x}{x} \cdot \frac{x ((x - 10) (x + 1))}{2 x (x + 1) (x + 6)}$

Rewrite the expression.

$(6 + x) \frac{(x - 10) (x + 1)}{2 x (x + 1) (x + 6)}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$(6 + x) \frac{(x - 10) (x + 1)}{2 x (x + 1) (x + 6)}$

Rewrite the expression.

$(6 + x) \frac{x - 10}{2 x (x + 6)}$

Multiply $6 + x$ by $\frac{x - 10}{2 x (x + 6)}$ .

$\frac{(6 + x) (x - 10)}{2 x (x + 6)}$

Cancel the common factor of $6 + x$ and $x + 6$ .

Reorder terms.

$\frac{(x + 6) (x - 10)}{2 x (x + 6)}$

Cancel the common factor.

$\frac{(x + 6) (x - 10)}{2 x (x + 6)}$

Rewrite the expression.

$\frac{x - 10}{2 x}$

Simplify ((36-x^2)/(6-x)*(x^2-2x-48)/(x^3-2x^2-48x))÷((2x^3+14x^2+12x)/(x^3-9x^2-10x))

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