Simplify ((3x^2-11x+6)/(x^2-9))÷((3x^2+x-2)/(3x^2+9x))

$\frac{3 x^{2} - 11 x + 6}{x^{2} - 9} \div \frac{3 x^{2} + x - 2}{3 x^{2} + 9 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 11 x + 6}{x^{2} - 9} \cdot \frac{3 x^{2} + 9 x}{3 x^{2} + x - 2}$

Step 2

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 $a \cdot c = 3 \cdot 6 = 18$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$\frac{3 x^{2} - 11 (x) + 6}{x^{2} - 9} \cdot \frac{3 x^{2} + 9 x}{3 x^{2} + x - 2}$

Rewrite $- 11$ as $- 2$ plus $- 9$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 3

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

Step 4

Simplify terms.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

Step 5

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 6

Cancel the common factor of $3 x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Cancel the common factor of $x + 3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 x}{x + 1}$

Simplify ((3x^2-11x+6)/(x^2-9))÷((3x^2+x-2)/(3x^2+9x))

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