Simplify (x+10)/(x^2+7x-18)*(3x^2-12x+12)/(3x+30)

$\frac{x + 10}{x^{2} + 7 x - 18} \cdot \frac{3 x^{2} - 12 x + 12}{3 x + 30}$

Factor $x^{2} + 7 x - 18$ 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 $- 18$ and whose sum is $7$ .

$- 2, 9$

Write the factored form using these integers.

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 x^{2} - 12 x + 12}{3 x + 30}$

Simplify the numerator.

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

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

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 (x^{2}) - 12 x + 12}{3 x + 30}$

Factor $3$ out of $- 12 x$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 (x^{2}) + 3 (- 4 x) + 12}{3 x + 30}$

Factor $3$ out of $12$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 x^{2} + 3 (- 4 x) + 3 \cdot 4}{3 x + 30}$

Factor $3$ out of $3 x^{2} + 3 (- 4 x)$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 (x^{2} - 4 x) + 3 \cdot 4}{3 x + 30}$

Factor $3$ out of $3 (x^{2} - 4 x) + 3 \cdot 4$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 (x^{2} - 4 x + 4)}{3 x + 30}$

Factor using the perfect square rule.

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

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 (x^{2} - 4 x + 2^{2})}{3 x + 30}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 (x^{2} - 2 \cdot x \cdot 2 + 2^{2})}{3 x + 30}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{3 x + 30}$

Simplify terms.

Factor $3$ out of $3 x + 30$ .

Factor $3$ out of $3 x$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{3 (x) + 30}$

Factor $3$ out of $30$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{3 x + 3 \cdot 10}$

Factor $3$ out of $3 x + 3 \cdot 10$ .

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{3 (x + 10)}$

Cancel the common factor of $x + 10$ .

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

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{(x + 10) \cdot 3}$

Cancel the common factor.

$\frac{x + 10}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{(x + 10) \cdot 3}$

Rewrite the expression.

$\frac{1}{(x - 2) (x + 9)} \cdot \frac{3 {(x - 2)}^{2}}{3}$

Cancel the common factor of $x - 2$ .

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

$\frac{1}{(x - 2) (x + 9)} \cdot \frac{(x - 2) (3 (x - 2))}{3}$

Cancel the common factor.

$\frac{1}{(x - 2) (x + 9)} \cdot \frac{(x - 2) (3 (x - 2))}{3}$

Rewrite the expression.

$\frac{1}{x + 9} \cdot \frac{3 (x - 2)}{3}$

Multiply $\frac{1}{x + 9}$ by $\frac{3 (x - 2)}{3}$ .

$\frac{3 (x - 2)}{(x + 9) \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x - 2)}{(x + 9) \cdot 3}$

Rewrite the expression.

$\frac{x - 2}{x + 9}$

Simplify (x+10)/(x^2+7x-18)*(3x^2-12x+12)/(3x+30)

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