Simplify ((9x^2+27x+18)/(x^2+8x+12)*(x+6)/(x+1))÷((x^2+7x+6)/(x^2+7x+6))

$\frac{9 x^{2} + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1} \div \frac{x^{2} + 7 x + 6}{x^{2} + 7 x + 6}$

To divide by a fraction, multiply by its reciprocal.

$\frac{9 x^{2} + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1} \frac{x^{2} + 7 x + 6}{x^{2} + 7 x + 6}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x^{2} + 7 x + 6$ .

Cancel the common factor.

$\frac{9 x^{2} + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1} \frac{x^{2} + 7 x + 6}{x^{2} + 7 x + 6}$

Rewrite the expression.

$\frac{9 x^{2} + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1} \cdot 1$

Multiply $\frac{9 x^{2} + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$ by $1$ .

$\frac{9 x^{2} + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

Simplify the numerator.

Factor $9$ out of $9 x^{2} + 27 x + 18$ .

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

$\frac{9 (x^{2}) + 27 x + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

Factor $9$ out of $27 x$ .

$\frac{9 (x^{2}) + 9 (3 x) + 18}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

Factor $9$ out of $18$ .

$\frac{9 x^{2} + 9 (3 x) + 9 \cdot 2}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

Factor $9$ out of $9 x^{2} + 9 (3 x)$ .

$\frac{9 (x^{2} + 3 x) + 9 \cdot 2}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

Factor $9$ out of $9 (x^{2} + 3 x) + 9 \cdot 2$ .

$\frac{9 (x^{2} + 3 x + 2)}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$\frac{9 ((x + 1) (x + 2))}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

$\frac{9 (x + 1) (x + 2)}{x^{2} + 8 x + 12} \cdot \frac{x + 6}{x + 1}$

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 $8$ .

$2, 6$

Write the factored form using these integers.

$\frac{9 (x + 1) (x + 2)}{(x + 2) (x + 6)} \cdot \frac{x + 6}{x + 1}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $9 (x + 1) (x + 2)$ .

$\frac{(x + 1) (9 (x + 2))}{(x + 2) (x + 6)} \cdot \frac{x + 6}{x + 1}$

Cancel the common factor.

$\frac{(x + 1) (9 (x + 2))}{(x + 2) (x + 6)} \cdot \frac{x + 6}{x + 1}$

Rewrite the expression.

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

Cancel the common factor of $x + 6$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{9 (x + 2)}{x + 2}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{9 (x + 2)}{x + 2}$

Divide $9$ by $1$ .

$9$

Simplify ((9x^2+27x+18)/(x^2+8x+12)*(x+6)/(x+1))÷((x^2+7x+6)/(x^2+7x+6))

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