Simplify (10x^2-640)/(x^2-17x+72)*(x^4-7x^3-18x^2)/(10(x+8)^2)

$\frac{10 x^{2} - 640}{x^{2} - 17 x + 72} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Simplify the numerator.

Factor $10$ out of $10 x^{2} - 640$ .

Factor $10$ out of $10 x^{2}$ .

$\frac{10 (x^{2}) - 640}{x^{2} - 17 x + 72} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Factor $10$ out of $- 640$ .

$\frac{10 x^{2} + 10 \cdot - 64}{x^{2} - 17 x + 72} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Factor $10$ out of $10 x^{2} + 10 \cdot - 64$ .

$\frac{10 (x^{2} - 64)}{x^{2} - 17 x + 72} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Rewrite $64$ as $8^{2}$ .

$\frac{10 (x^{2} - 8^{2})}{x^{2} - 17 x + 72} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

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 $b = 8$ .

$\frac{10 (x + 8) (x - 8)}{x^{2} - 17 x + 72} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Factor $x^{2} - 17 x + 72$ 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 $72$ and whose sum is $- 17$ .

$- 9, - 8$

Write the factored form using these integers.

$\frac{10 (x + 8) (x - 8)}{(x - 9) (x - 8)} \cdot \frac{x^{4} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Simplify the numerator.

Factor $x^{2}$ out of $x^{4} - 7 x^{3} - 18 x^{2}$ .

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

$\frac{10 (x + 8) (x - 8)}{(x - 9) (x - 8)} \cdot \frac{x^{2} x^{2} - 7 x^{3} - 18 x^{2}}{10 {(x + 8)}^{2}}$

Factor $x^{2}$ out of $- 7 x^{3}$ .

$\frac{10 (x + 8) (x - 8)}{(x - 9) (x - 8)} \cdot \frac{x^{2} x^{2} + x^{2} (- 7 x) - 18 x^{2}}{10 {(x + 8)}^{2}}$

Factor $x^{2}$ out of $- 18 x^{2}$ .

$\frac{10 (x + 8) (x - 8)}{(x - 9) (x - 8)} \cdot \frac{x^{2} x^{2} + x^{2} (- 7 x) + x^{2} \cdot - 18}{10 {(x + 8)}^{2}}$

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

$\frac{10 (x + 8) (x - 8)}{(x - 9) (x - 8)} \cdot \frac{x^{2} (x^{2} - 7 x) + x^{2} \cdot - 18}{10 {(x + 8)}^{2}}$

Factor $x^{2}$ out of $x^{2} (x^{2} - 7 x) + x^{2} \cdot - 18$ .

$\frac{10 (x + 8) (x - 8)}{(x - 9) (x - 8)} \cdot \frac{x^{2} (x^{2} - 7 x - 18)}{10 {(x + 8)}^{2}}$

Factor $x^{2} - 7 x - 18$ 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 $- 7$ .

$- 9, 2$

Write the factored form using these integers.

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

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

Simplify terms.

Combine.

$\frac{10 (x + 8) (x - 8) (x^{2} (x - 9) (x + 2))}{(x - 9) (x - 8) (10 {(x + 8)}^{2})}$

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 (x + 8) (x - 8) (x^{2} (x - 9) (x + 2))}{(x - 9) (x - 8) (10 {(x + 8)}^{2})}$

Rewrite the expression.

$\frac{((x + 8) (x - 8)) (x^{2} (x - 9) (x + 2))}{(x - 9) (x - 8) ({(x + 8)}^{2})}$

Cancel the common factor of $x + 8$ and ${(x + 8)}^{2}$ .

Factor $x + 8$ out of $((x + 8) (x - 8)) (x^{2} (x - 9) (x + 2))$ .

$\frac{(x + 8) ((x - 8) (x^{2} (x - 9) (x + 2)))}{(x - 9) (x - 8) ({(x + 8)}^{2})}$

Cancel the common factors.

Factor $x + 8$ out of $(x - 9) (x - 8) ({(x + 8)}^{2})$ .

$\frac{(x + 8) ((x - 8) (x^{2} (x - 9) (x + 2)))}{(x + 8) ((x - 9) (x - 8) (x + 8))}$

Cancel the common factor.

$\frac{(x + 8) ((x - 8) (x^{2} (x - 9) (x + 2)))}{(x + 8) ((x - 9) (x - 8) (x + 8))}$

Rewrite the expression.

$\frac{(x - 8) (x^{2} (x - 9) (x + 2))}{(x - 9) (x - 8) (x + 8)}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{(x - 8) (x^{2} (x - 9) (x + 2))}{(x - 9) (x - 8) (x + 8)}$

Rewrite the expression.

$\frac{x^{2} (x - 9) (x + 2)}{(x - 9) (x + 8)}$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{x^{2} (x - 9) (x + 2)}{(x - 9) (x + 8)}$

Rewrite the expression.

$\frac{x^{2} (x + 2)}{x + 8}$

Simplify (10x^2-640)/(x^2-17x+72)*(x^4-7x^3-18x^2)/(10(x+8)^2)

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