Simplify (x+3)/(5x+20)*(x^2+3x+2)/(x^2+5x+6)

$\frac{x + 3}{5 x + 20} \cdot \frac{x^{2} + 3 x + 2}{x^{2} + 5 x + 6}$

Step 1

Factor $5$ out of $5 x + 20$ .

Factor $5$ out of $5 x$ .

$\frac{x + 3}{5 (x) + 20} \cdot \frac{x^{2} + 3 x + 2}{x^{2} + 5 x + 6}$

Factor $5$ out of $20$ .

$\frac{x + 3}{5 x + 5 \cdot 4} \cdot \frac{x^{2} + 3 x + 2}{x^{2} + 5 x + 6}$

Factor $5$ out of $5 x + 5 \cdot 4$ .

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

Step 2

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.

$\frac{x + 3}{5 (x + 4)} \cdot \frac{(x + 1) (x + 2)}{x^{2} + 5 x + 6}$

Step 3

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 $5$ .

$2, 3$

Write the factored form using these integers.

$\frac{x + 3}{5 (x + 4)} \cdot \frac{(x + 1) (x + 2)}{(x + 2) (x + 3)}$

Step 4

Simplify terms.

Cancel the common factor of $x + 3$ .

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

$\frac{x + 3}{5 (x + 4)} \cdot \frac{(x + 1) (x + 2)}{(x + 3) (x + 2)}$

Cancel the common factor.

$\frac{x + 3}{5 (x + 4)} \cdot \frac{(x + 1) (x + 2)}{(x + 3) (x + 2)}$

Rewrite the expression.

$\frac{1}{5 (x + 4)} \cdot \frac{(x + 1) (x + 2)}{x + 2}$

Multiply $\frac{1}{5 (x + 4)}$ by $\frac{(x + 1) (x + 2)}{x + 2}$ .

$\frac{(x + 1) (x + 2)}{5 (x + 4) (x + 2)}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{(x + 1) (x + 2)}{5 (x + 4) (x + 2)}$

Rewrite the expression.

$\frac{x + 1}{5 (x + 4)}$

Simplify (x+3)/(5x+20)*(x^2+3x+2)/(x^2+5x+6)

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