Simplify (x^5-x^4-6x^3)/(4x+8)*(2x^2-50)/(x^2-8x+15)

$\frac{x^{5} - x^{4} - 6 x^{3}}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Simplify the numerator.

Factor $x^{3}$ out of $x^{5} - x^{4} - 6 x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

$\frac{x^{3} x^{2} - x^{4} - 6 x^{3}}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

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

$\frac{x^{3} x^{2} + x^{3} (- x) - 6 x^{3}}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Factor $x^{3}$ out of $- 6 x^{3}$ .

$\frac{x^{3} x^{2} + x^{3} (- x) + x^{3} \cdot - 6}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

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

$\frac{x^{3} (x^{2} - x) + x^{3} \cdot - 6}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Factor $x^{3}$ out of $x^{3} (x^{2} - x) + x^{3} \cdot - 6$ .

$\frac{x^{3} (x^{2} - x - 6)}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Factor $x^{2} - x - 6$ 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 $- 1$ .

$- 3, 2$

Write the factored form using these integers.

$\frac{x^{3} ((x - 3) (x + 2))}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

$\frac{x^{3} (x - 3) (x + 2)}{4 x + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Factor $4$ out of $4 x + 8$ .

Factor $4$ out of $4 x$ .

$\frac{x^{3} (x - 3) (x + 2)}{4 (x) + 8} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Factor $4$ out of $8$ .

$\frac{x^{3} (x - 3) (x + 2)}{4 x + 4 \cdot 2} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Factor $4$ out of $4 x + 4 \cdot 2$ .

$\frac{x^{3} (x - 3) (x + 2)}{4 (x + 2)} \cdot \frac{2 x^{2} - 50}{x^{2} - 8 x + 15}$

Simplify the numerator.

Factor $2$ out of $2 x^{2} - 50$ .

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

$\frac{x^{3} (x - 3) (x + 2)}{4 (x + 2)} \cdot \frac{2 (x^{2}) - 50}{x^{2} - 8 x + 15}$

Factor $2$ out of $- 50$ .

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

Factor $2$ out of $2 x^{2} + 2 \cdot - 25$ .

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

Rewrite $25$ as $5^{2}$ .

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

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 $b = 5$ .

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

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

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

Simplify terms.

Combine.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $(x^{3}) (2 (x + 5) (x - 5))$ .

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

Cancel the common factors.

Factor $2$ out of $4 (x - 5)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Simplify (x^5-x^4-6x^3)/(4x+8)*(2x^2-50)/(x^2-8x+15)

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