Simplify (6x^5-66x^4+144x^3)/(x^2-5x+6)*(x^2+2x-8)/(x^2-4x-32)

$\frac{6 x^{5} - 66 x^{4} + 144 x^{3}}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Step 1

Simplify the numerator.

Factor $6 x^{3}$ out of $6 x^{5} - 66 x^{4} + 144 x^{3}$ .

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

$\frac{6 x^{3} (x^{2}) - 66 x^{4} + 144 x^{3}}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Factor $6 x^{3}$ out of $- 66 x^{4}$ .

$\frac{6 x^{3} (x^{2}) + 6 x^{3} (- 11 x) + 144 x^{3}}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Factor $6 x^{3}$ out of $144 x^{3}$ .

$\frac{6 x^{3} (x^{2}) + 6 x^{3} (- 11 x) + 6 x^{3} (24)}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Factor $6 x^{3}$ out of $6 x^{3} (x^{2}) + 6 x^{3} (- 11 x)$ .

$\frac{6 x^{3} (x^{2} - 11 x) + 6 x^{3} (24)}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Factor $6 x^{3}$ out of $6 x^{3} (x^{2} - 11 x) + 6 x^{3} (24)$ .

$\frac{6 x^{3} (x^{2} - 11 x + 24)}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Factor $x^{2} - 11 x + 24$ 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 $24$ and whose sum is $- 11$ .

$- 8, - 3$

Write the factored form using these integers.

$\frac{6 x^{3} ((x - 8) (x - 3))}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

$\frac{6 x^{3} (x - 8) (x - 3)}{x^{2} - 5 x + 6} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Step 2

Factor $x^{2} - 5 x + 6$ 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 $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

$\frac{6 x^{3} (x - 8) (x - 3)}{(x - 3) (x - 2)} \cdot \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 32}$

Step 3

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

$- 2, 4$

Write the factored form using these integers.

$\frac{6 x^{3} (x - 8) (x - 3)}{(x - 3) (x - 2)} \cdot \frac{(x - 2) (x + 4)}{x^{2} - 4 x - 32}$

Step 4

Factor $x^{2} - 4 x - 32$ 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 $- 32$ and whose sum is $- 4$ .

$- 8, 4$

Write the factored form using these integers.

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

Step 5

Simplify terms.

Combine.

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

Cancel the common factor of $x - 8$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(6 x^{3} (x - 3)) ((x - 2) (x + 4))}{(x - 3) (x - 2) (x + 4)}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{6 x^{3} (x - 3) ((x - 2) (x + 4))}{(x - 3) (x - 2) (x + 4)}$

Rewrite the expression.

$\frac{(6 x^{3}) ((x - 2) (x + 4))}{(x - 2) (x + 4)}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{6 x^{3} ((x - 2) (x + 4))}{(x - 2) (x + 4)}$

Rewrite the expression.

$\frac{(6 x^{3}) (x + 4)}{x + 4}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{6 x^{3} (x + 4)}{x + 4}$

Divide $6 x^{3}$ by $1$ .

$6 x^{3}$

Simplify (6x^5-66x^4+144x^3)/(x^2-5x+6)*(x^2+2x-8)/(x^2-4x-32)

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