Simplify (3x^2-9x-12)/(6x^2+9x+3)

$\frac{3 x^{2} - 9 x - 12}{6 x^{2} + 9 x + 3}$

Step 1

Cancel the common factor of $3 x^{2} - 9 x - 12$ and $6 x^{2} + 9 x + 3$ .

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

$\frac{3 (x^{2}) - 9 x - 12}{6 x^{2} + 9 x + 3}$

Factor $3$ out of $- 9 x$ .

$\frac{3 (x^{2}) + 3 (- 3 x) - 12}{6 x^{2} + 9 x + 3}$

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

$\frac{3 (x^{2} - 3 x) - 12}{6 x^{2} + 9 x + 3}$

Factor $3$ out of $- 12$ .

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

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

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

Cancel the common factors.

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

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

Factor $3$ out of $9 x$ .

$\frac{3 (x^{2} - 3 x - 4)}{3 (2 x^{2}) + 3 (3 x) + 3}$

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

$\frac{3 (x^{2} - 3 x - 4)}{3 (2 x^{2} + 3 x) + 3}$

Factor $3$ out of $3$ .

$\frac{3 (x^{2} - 3 x - 4)}{3 (2 x^{2} + 3 x) + 3 (1)}$

Factor $3$ out of $3 (2 x^{2} + 3 x) + 3 (1)$ .

$\frac{3 (x^{2} - 3 x - 4)}{3 (2 x^{2} + 3 x + 1)}$

Cancel the common factor.

$\frac{3 (x^{2} - 3 x - 4)}{3 (2 x^{2} + 3 x + 1)}$

Rewrite the expression.

$\frac{x^{2} - 3 x - 4}{2 x^{2} + 3 x + 1}$

Step 2

Factor $x^{2} - 3 x - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

$\frac{(x - 4) (x + 1)}{2 x^{2} + 3 x + 1}$

Step 3

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

$\frac{(x - 4) (x + 1)}{2 x^{2} + 3 (x) + 1}$

Rewrite $3$ as $1$ plus $2$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 4

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 4}{2 x + 1}$

Simplify (3x^2-9x-12)/(6x^2+9x+3)

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