Simplify (3x^2+6x)/(x^2-8x+12)*(x^2+8x-20)/(x^4+12x^3+20x^2)

$\frac{3 x^{2} + 6 x}{x^{2} - 8 x + 12} \cdot \frac{x^{2} + 8 x - 20}{x^{4} + 12 x^{3} + 20 x^{2}}$

Step 1

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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

$\frac{3 x (x) + 6 x}{x^{2} - 8 x + 12} \cdot \frac{x^{2} + 8 x - 20}{x^{4} + 12 x^{3} + 20 x^{2}}$

Factor $3 x$ out of $6 x$ .

$\frac{3 x (x) + 3 x (2)}{x^{2} - 8 x + 12} \cdot \frac{x^{2} + 8 x - 20}{x^{4} + 12 x^{3} + 20 x^{2}}$

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

$\frac{3 x (x + 2)}{x^{2} - 8 x + 12} \cdot \frac{x^{2} + 8 x - 20}{x^{4} + 12 x^{3} + 20 x^{2}}$

Step 2

Factor $x^{2} - 8 x + 12$ 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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Write the factored form using these integers.

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{x^{2} + 8 x - 20}{x^{4} + 12 x^{3} + 20 x^{2}}$

Step 3

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.

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{4} + 12 x^{3} + 20 x^{2}}$

Step 4

Simplify the denominator.

Factor $x^{2}$ out of $x^{4} + 12 x^{3} + 20 x^{2}$ .

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

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} x^{2} + 12 x^{3} + 20 x^{2}}$

Factor $x^{2}$ out of $12 x^{3}$ .

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} x^{2} + x^{2} (12 x) + 20 x^{2}}$

Factor $x^{2}$ out of $20 x^{2}$ .

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} x^{2} + x^{2} (12 x) + x^{2} \cdot 20}$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (12 x)$ .

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} (x^{2} + 12 x) + x^{2} \cdot 20}$

Factor $x^{2}$ out of $x^{2} (x^{2} + 12 x) + x^{2} \cdot 20$ .

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} (x^{2} + 12 x + 20)}$

Factor $x^{2} + 12 x + 20$ 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 $12$ .

$2, 10$

Write the factored form using these integers.

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} ((x + 2) (x + 10))}$

$\frac{3 x (x + 2)}{(x - 6) (x - 2)} \cdot \frac{(x - 2) (x + 10)}{x^{2} (x + 2) (x + 10)}$

Step 5

Simplify terms.

Combine.

$\frac{3 x (x + 2) ((x - 2) (x + 10))}{(x - 6) (x - 2) (x^{2} (x + 2) (x + 10))}$

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

Factor $x$ out of $3 x (x + 2) ((x - 2) (x + 10))$ .

$\frac{x ((3 (x + 2)) ((x - 2) (x + 10)))}{(x - 6) (x - 2) (x^{2} (x + 2) (x + 10))}$

Cancel the common factors.

Factor $x$ out of $(x - 6) (x - 2) (x^{2} (x + 2) (x + 10))$ .

$\frac{x ((3 (x + 2)) ((x - 2) (x + 10)))}{x ((x - 6) (x - 2) ((x (x + 2)) (x + 10)))}$

Cancel the common factor.

$\frac{x ((3 (x + 2)) ((x - 2) (x + 10)))}{x ((x - 6) (x - 2) ((x (x + 2)) (x + 10)))}$

Rewrite the expression.

$\frac{(3 (x + 2)) ((x - 2) (x + 10))}{(x - 6) (x - 2) ((x (x + 2)) (x + 10))}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{3 (x + 2) ((x - 2) (x + 10))}{(x - 6) (x - 2) (x (x + 2) (x + 10))}$

Rewrite the expression.

$\frac{(3) ((x - 2) (x + 10))}{(x - 6) (x - 2) ((x) (x + 10))}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{3 ((x - 2) (x + 10))}{(x - 6) (x - 2) ((x) (x + 10))}$

Rewrite the expression.

$\frac{(3) (x + 10)}{(x - 6) ((x) (x + 10))}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{3 (x + 10)}{(x - 6) (x (x + 10))}$

Rewrite the expression.

$\frac{3}{(x - 6) (x)}$

Reorder factors in $\frac{3}{(x - 6) (x)}$ .

$\frac{3}{x (x - 6)}$

Simplify (3x^2+6x)/(x^2-8x+12)*(x^2+8x-20)/(x^4+12x^3+20x^2)

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