Simplify ((x^2-9)/(3x^2+8x-3))÷((x-3)/(x+2))

$\frac{x^{2} - 9}{3 x^{2} + 8 x - 3} \div \frac{x - 3}{x + 2}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 9}{3 x^{2} + 8 x - 3} \cdot \frac{x + 2}{x - 3}$

Simplify the numerator.

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

$\frac{x^{2} - 3^{2}}{3 x^{2} + 8 x - 3} \cdot \frac{x + 2}{x - 3}$

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 $b = 3$ .

$\frac{(x + 3) (x - 3)}{3 x^{2} + 8 x - 3} \cdot \frac{x + 2}{x - 3}$

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = 8$ .

Factor $8$ out of $8 x$ .

$\frac{(x + 3) (x - 3)}{3 x^{2} + 8 (x) - 3} \cdot \frac{x + 2}{x - 3}$

Rewrite $8$ as $- 1$ plus $9$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

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

Cancel the common factor of $x - 3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify ((x^2-9)/(3x^2+8x-3))÷((x-3)/(x+2))

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