Simplify (x^2-64)/(5(x+7)^2)*(x^2-49)/(x^4-15x^3+56x^2)

$\frac{x^{2} - 64}{5 {(x + 7)}^{2}} \cdot \frac{x^{2} - 49}{x^{4} - 15 x^{3} + 56 x^{2}}$

Simplify the numerator.

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

$\frac{x^{2} - 8^{2}}{5 {(x + 7)}^{2}} \cdot \frac{x^{2} - 49}{x^{4} - 15 x^{3} + 56 x^{2}}$

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 $b = 8$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{x^{2} - 49}{x^{4} - 15 x^{3} + 56 x^{2}}$

Simplify the numerator.

Rewrite $49$ as $7^{2}$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{x^{2} - 7^{2}}{x^{4} - 15 x^{3} + 56 x^{2}}$

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 $b = 7$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{4} - 15 x^{3} + 56 x^{2}}$

Simplify the denominator.

Factor $x^{2}$ out of $x^{4} - 15 x^{3} + 56 x^{2}$ .

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

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} x^{2} - 15 x^{3} + 56 x^{2}}$

Factor $x^{2}$ out of $- 15 x^{3}$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} x^{2} + x^{2} (- 15 x) + 56 x^{2}}$

Factor $x^{2}$ out of $56 x^{2}$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} x^{2} + x^{2} (- 15 x) + x^{2} \cdot 56}$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 15 x)$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} (x^{2} - 15 x) + x^{2} \cdot 56}$

Factor $x^{2}$ out of $x^{2} (x^{2} - 15 x) + x^{2} \cdot 56$ .

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} (x^{2} - 15 x + 56)}$

Factor $x^{2} - 15 x + 56$ 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 $56$ and whose sum is $- 15$ .

$- 8, - 7$

Write the factored form using these integers.

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} ((x - 8) (x - 7))}$

$\frac{(x + 8) (x - 8)}{5 {(x + 7)}^{2}} \cdot \frac{(x + 7) (x - 7)}{x^{2} (x - 8) (x - 7)}$

Simplify terms.

Combine.

$\frac{(x + 8) (x - 8) ((x + 7) (x - 7))}{5 {(x + 7)}^{2} (x^{2} (x - 8) (x - 7))}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{(x + 8) (x - 8) ((x + 7) (x - 7))}{5 {(x + 7)}^{2} (x^{2} (x - 8) (x - 7))}$

Rewrite the expression.

$\frac{(x + 8) ((x + 7) (x - 7))}{5 {(x + 7)}^{2} (x^{2} (x - 7))}$

Cancel the common factor of $x + 7$ and ${(x + 7)}^{2}$ .

Factor $x + 7$ out of $(x + 8) ((x + 7) (x - 7))$ .

$\frac{(x + 7) ((x + 8) (x - 7))}{5 {(x + 7)}^{2} (x^{2} (x - 7))}$

Cancel the common factors.

Factor $x + 7$ out of $5 {(x + 7)}^{2} (x^{2} (x - 7))$ .

$\frac{(x + 7) ((x + 8) (x - 7))}{(x + 7) (5 (x + 7) (x^{2} (x - 7)))}$

Cancel the common factor.

$\frac{(x + 7) ((x + 8) (x - 7))}{(x + 7) (5 (x + 7) (x^{2} (x - 7)))}$

Rewrite the expression.

$\frac{(x + 8) (x - 7)}{5 (x + 7) (x^{2} (x - 7))}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{(x + 8) (x - 7)}{5 (x + 7) (x^{2} (x - 7))}$

Rewrite the expression.

$\frac{x + 8}{5 (x + 7) (x^{2})}$

Reorder factors in $\frac{x + 8}{5 (x + 7) (x^{2})}$ .

$\frac{x + 8}{5 (x^{2}) (x + 7)}$

Simplify (x^2-64)/(5(x+7)^2)*(x^2-49)/(x^4-15x^3+56x^2)

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