Simplify (x^2+8x-20)/(x^4+7x^3-18x^2)*(4x^4+8x^3-252x^2)/(6x+60)

$\frac{x^{2} + 8 x - 20}{x^{4} + 7 x^{3} - 18 x^{2}} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

Step 1

Factor $x^{2} + 8 x - 20$ 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 $- 20$ and whose sum is $8$ .

$- 2, 10$

Write the factored form using these integers.

$\frac{(x - 2) (x + 10)}{x^{4} + 7 x^{3} - 18 x^{2}} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

Step 2

Simplify the denominator.

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

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

$\frac{(x - 2) (x + 10)}{x^{2} x^{2} + 7 x^{3} - 18 x^{2}} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

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

$\frac{(x - 2) (x + 10)}{x^{2} x^{2} + x^{2} (7 x) - 18 x^{2}} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

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

$\frac{(x - 2) (x + 10)}{x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot - 18} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

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

$\frac{(x - 2) (x + 10)}{x^{2} (x^{2} + 7 x) + x^{2} \cdot - 18} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

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

$\frac{(x - 2) (x + 10)}{x^{2} (x^{2} + 7 x - 18)} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

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 $7$ .

$- 2, 9$

Write the factored form using these integers.

$\frac{(x - 2) (x + 10)}{x^{2} ((x - 2) (x + 9))} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{4} + 8 x^{3} - 252 x^{2}}{6 x + 60}$

Step 3

Simplify the numerator.

Factor $4 x^{2}$ out of $4 x^{4} + 8 x^{3} - 252 x^{2}$ .

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

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x^{2}) + 8 x^{3} - 252 x^{2}}{6 x + 60}$

Factor $4 x^{2}$ out of $8 x^{3}$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x^{2}) + 4 x^{2} (2 x) - 252 x^{2}}{6 x + 60}$

Factor $4 x^{2}$ out of $- 252 x^{2}$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x^{2}) + 4 x^{2} (2 x) + 4 x^{2} (- 63)}{6 x + 60}$

Factor $4 x^{2}$ out of $4 x^{2} (x^{2}) + 4 x^{2} (2 x)$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x^{2} + 2 x) + 4 x^{2} (- 63)}{6 x + 60}$

Factor $4 x^{2}$ out of $4 x^{2} (x^{2} + 2 x) + 4 x^{2} (- 63)$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x^{2} + 2 x - 63)}{6 x + 60}$

Factor $x^{2} + 2 x - 63$ 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 $- 63$ and whose sum is $2$ .

$- 7, 9$

Write the factored form using these integers.

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} ((x - 7) (x + 9))}{6 x + 60}$

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x - 7) (x + 9)}{6 x + 60}$

Step 4

Simplify terms.

Factor $6$ out of $6 x + 60$ .

Factor $6$ out of $6 x$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x - 7) (x + 9)}{6 (x) + 60}$

Factor $6$ out of $60$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x - 7) (x + 9)}{6 x + 6 \cdot 10}$

Factor $6$ out of $6 x + 6 \cdot 10$ .

$\frac{(x - 2) (x + 10)}{x^{2} (x - 2) (x + 9)} \cdot \frac{4 x^{2} (x - 7) (x + 9)}{6 (x + 10)}$

Combine.

$\frac{(x - 2) (x + 10) (4 x^{2} (x - 7) (x + 9))}{x^{2} (x - 2) (x + 9) (6 (x + 10))}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{(x - 2) (x + 10) (4 x^{2} (x - 7) (x + 9))}{x^{2} (x - 2) (x + 9) (6 (x + 10))}$

Rewrite the expression.

$\frac{(x + 10) (4 x^{2} (x - 7) (x + 9))}{(x^{2} (x + 9)) (6 (x + 10))}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{(x + 10) (4 x^{2} (x - 7) (x + 9))}{x^{2} (x + 9) (6 (x + 10))}$

Rewrite the expression.

$\frac{4 x^{2} (x - 7) (x + 9)}{(x^{2} (x + 9)) \cdot (6)}$

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4 x^{2} (x - 7) (x + 9)$ .

$\frac{2 (2 x^{2} (x - 7) (x + 9))}{(x^{2} (x + 9)) \cdot (6)}$

Cancel the common factors.

Factor $2$ out of $(x^{2} (x + 9)) \cdot (6)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(2 (x - 7)) (x + 9)}{(x + 9) \cdot (3)}$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$\frac{2 (x - 7) (x + 9)}{(x + 9) \cdot 3}$

Rewrite the expression.

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

Simplify (x^2+8x-20)/(x^4+7x^3-18x^2)*(4x^4+8x^3-252x^2)/(6x+60)

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