Simplify ((9x^3+81x^2+126x)/(x^2-4)*4/(6x^2+12x))÷((x^2+9x+14)/(x^2-4))

$\frac{9 x^{3} + 81 x^{2} + 126 x}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \div \frac{x^{2} + 9 x + 14}{x^{2} - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9 x^{3} + 81 x^{2} + 126 x}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Step 2

Simplify the numerator.

Factor $9 x$ out of $9 x^{3} + 81 x^{2} + 126 x$ .

Factor $9 x$ out of $9 x^{3}$ .

$\frac{9 x (x^{2}) + 81 x^{2} + 126 x}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $9 x$ out of $81 x^{2}$ .

$\frac{9 x (x^{2}) + 9 x (9 x) + 126 x}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $9 x$ out of $126 x$ .

$\frac{9 x (x^{2}) + 9 x (9 x) + 9 x (14)}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

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

$\frac{9 x (x^{2} + 9 x) + 9 x (14)}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $9 x$ out of $9 x (x^{2} + 9 x) + 9 x (14)$ .

$\frac{9 x (x^{2} + 9 x + 14)}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $x^{2} + 9 x + 14$ 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 $14$ and whose sum is $9$ .

$2, 7$

Write the factored form using these integers.

$\frac{9 x ((x + 2) (x + 7))}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

$\frac{9 x (x + 2) (x + 7)}{x^{2} - 4} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Step 3

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{9 x (x + 2) (x + 7)}{x^{2} - 2^{2}} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

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 $b = 2$ .

$\frac{9 x (x + 2) (x + 7)}{(x + 2) (x - 2)} \cdot \frac{4}{6 x^{2} + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Step 4

Simplify terms.

Factor $6 x$ out of $6 x^{2} + 12 x$ .

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

$\frac{9 x (x + 2) (x + 7)}{(x + 2) (x - 2)} \cdot \frac{4}{6 x (x) + 12 x} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $6 x$ out of $12 x$ .

$\frac{9 x (x + 2) (x + 7)}{(x + 2) (x - 2)} \cdot \frac{4}{6 x (x) + 6 x (2)} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $6 x$ out of $6 x (x) + 6 x (2)$ .

$\frac{9 x (x + 2) (x + 7)}{(x + 2) (x - 2)} \cdot \frac{4}{6 x (x + 2)} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Cancel the common factor of $3 x (x + 2)$ .

Factor $3 x (x + 2)$ out of $9 x (x + 2) (x + 7)$ .

$\frac{3 x (x + 2) ((3) (x + 7))}{(x + 2) (x - 2)} \cdot \frac{4}{6 x (x + 2)} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Factor $3 x (x + 2)$ out of $6 x (x + 2)$ .

$\frac{3 x (x + 2) ((3) (x + 7))}{(x + 2) (x - 2)} \cdot \frac{4}{3 x (x + 2) (2)} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Cancel the common factor.

$\frac{3 x (x + 2) ((3) (x + 7))}{(x + 2) (x - 2)} \cdot \frac{4}{3 x (x + 2) \cdot 2} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Rewrite the expression.

$\frac{(3) (x + 7)}{(x + 2) (x - 2)} \cdot \frac{4}{2} \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Multiply $\frac{(3) (x + 7)}{(x + 2) (x - 2)}$ by $\frac{4}{2}$ .

$\frac{(3) (x + 7) \cdot 4}{(x + 2) (x - 2) \cdot 2} \cdot \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Multiply $4$ by $3$ .

$\frac{12 (x + 7)}{(x + 2) (x - 2) \cdot 2} \cdot \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12 (x + 7)$ .

$\frac{2 (6 (x + 7))}{(x + 2) (x - 2) \cdot 2} \cdot \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Cancel the common factors.

Factor $2$ out of $(x + 2) (x - 2) \cdot 2$ .

$\frac{2 (6 (x + 7))}{2 \cdot ((x + 2) (x - 2))} \cdot \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Cancel the common factor.

$\frac{2 (6 (x + 7))}{2 \cdot ((x + 2) (x - 2))} \cdot \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Rewrite the expression.

$\frac{6 (x + 7)}{(x + 2) (x - 2)} \cdot \frac{x^{2} - 4}{x^{2} + 9 x + 14}$

Step 5

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{6 (x + 7)}{(x + 2) (x - 2)} \cdot \frac{x^{2} - 2^{2}}{x^{2} + 9 x + 14}$

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 $b = 2$ .

$\frac{6 (x + 7)}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{x^{2} + 9 x + 14}$

Step 6

Factor $x^{2} + 9 x + 14$ 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 $14$ and whose sum is $9$ .

$2, 7$

Write the factored form using these integers.

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

Step 7

Simplify terms.

Cancel the common factor of $x + 7$ .

Factor $x + 7$ out of $6 (x + 7)$ .

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

Factor $x + 7$ out of $(x + 2) (x + 7)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $(x + 2) (x - 2)$ .

Cancel the common factor.

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

Rewrite the expression.

$6 \frac{1}{x + 2}$

Combine $6$ and $\frac{1}{x + 2}$ .

$\frac{6}{x + 2}$

Simplify ((9x^3+81x^2+126x)/(x^2-4)*4/(6x^2+12x))÷((x^2+9x+14)/(x^2-4))

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