Simplify (6x^4-294x^2)/(x^4+15x^3+56x^2)*(x^2-64)/(2x^2-30x+112)

$\frac{6 x^{4} - 294 x^{2}}{x^{4} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

Simplify the numerator.

Factor $6 x^{2}$ out of $6 x^{4} - 294 x^{2}$ .

Factor $6 x^{2}$ out of $6 x^{4}$ .

$\frac{6 x^{2} (x^{2}) - 294 x^{2}}{x^{4} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

Factor $6 x^{2}$ out of $- 294 x^{2}$ .

$\frac{6 x^{2} (x^{2}) + 6 x^{2} (- 49)}{x^{4} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

Factor $6 x^{2}$ out of $6 x^{2} (x^{2}) + 6 x^{2} (- 49)$ .

$\frac{6 x^{2} (x^{2} - 49)}{x^{4} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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

$\frac{6 x^{2} (x^{2} - 7^{2})}{x^{4} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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 $b = 7$ .

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{4} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

Simplify the denominator.

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

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} x^{2} + 15 x^{3} + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} x^{2} + x^{2} (15 x) + 56 x^{2}} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} x^{2} + x^{2} (15 x) + x^{2} \cdot 56} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x^{2} + 15 x) + x^{2} \cdot 56} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x^{2} + 15 x + 56)} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

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 $15$ .

$7, 8$

Write the factored form using these integers.

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} ((x + 7) (x + 8))} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{x^{2} - 64}{2 x^{2} - 30 x + 112}$

Simplify the numerator.

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{x^{2} - 8^{2}}{2 x^{2} - 30 x + 112}$

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 $b = 8$ .

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 x^{2} - 30 x + 112}$

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 30 x + 112$ .

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 (x^{2}) - 30 x + 112}$

Factor $2$ out of $- 30 x$ .

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 (x^{2}) + 2 (- 15 x) + 112}$

Factor $2$ out of $112$ .

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 x^{2} + 2 (- 15 x) + 2 \cdot 56}$

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 (x^{2} - 15 x) + 2 \cdot 56}$

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

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 (x^{2} - 15 x + 56)}$

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.

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 ((x - 8) (x - 7))}$

$\frac{6 x^{2} (x + 7) (x - 7)}{x^{2} (x + 7) (x + 8)} \cdot \frac{(x + 8) (x - 8)}{2 (x - 8) (x - 7)}$

Simplify terms.

Combine.

$\frac{6 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8))}{x^{2} (x + 7) (x + 8) (2 (x - 8) (x - 7))}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8))$ .

$\frac{2 (3 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8)))}{x^{2} (x + 7) (x + 8) (2 (x - 8) (x - 7))}$

Cancel the common factors.

Factor $2$ out of $x^{2} (x + 7) (x + 8) (2 (x - 8) (x - 7))$ .

$\frac{2 (3 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8)))}{2 (x^{2} (x + 7) (x + 8) ((x - 8) (x - 7)))}$

Cancel the common factor.

$\frac{2 (3 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8)))}{2 (x^{2} (x + 7) (x + 8) ((x - 8) (x - 7)))}$

Rewrite the expression.

$\frac{3 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8))}{x^{2} (x + 7) (x + 8) ((x - 8) (x - 7))}$

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

Cancel the common factor.

$\frac{3 x^{2} (x + 7) (x - 7) ((x + 8) (x - 8))}{x^{2} (x + 7) (x + 8) ((x - 8) (x - 7))}$

Rewrite the expression.

$\frac{((3 (x + 7)) (x - 7)) ((x + 8) (x - 8))}{((x + 7) (x + 8)) ((x - 8) (x - 7))}$

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{3 (x + 7) (x - 7) ((x + 8) (x - 8))}{(x + 7) (x + 8) ((x - 8) (x - 7))}$

Rewrite the expression.

$\frac{((3) (x - 7)) ((x + 8) (x - 8))}{(x + 8) ((x - 8) (x - 7))}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{3 (x - 7) ((x + 8) (x - 8))}{(x + 8) ((x - 8) (x - 7))}$

Rewrite the expression.

$\frac{(3) ((x + 8) (x - 8))}{(x + 8) (x - 8)}$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{3 ((x + 8) (x - 8))}{(x + 8) (x - 8)}$

Rewrite the expression.

$\frac{(3) (x - 8)}{x - 8}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{3 (x - 8)}{x - 8}$

Divide $3$ by $1$ .

$3$

Simplify (6x^4-294x^2)/(x^4+15x^3+56x^2)*(x^2-64)/(2x^2-30x+112)

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