Divide ((c^2-4)/(c+3))÷((c+2)/(3(c^2-9)))

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 2$ .

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

Step 3

Cancel the common factor of $c + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Combine $3$ and $\frac{c - 2}{c + 3}$ .

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

Step 5

Multiply $\frac{3 (c - 2)}{c + 3}$ by $c^{2} - 9$ .

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

Step 6

Simplify the numerator.

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 3$ .

$\frac{3 (c - 2) (c + 3) (c - 3)}{c + 3}$

Step 7

Cancel the common factor of $c + 3$ .

Cancel the common factor.

$\frac{3 (c - 2) (c + 3) (c - 3)}{c + 3}$

Divide $3 (c - 2) (c - 3)$ by $1$ .

$3 (c - 2) (c - 3)$

Step 8

Apply the distributive property.

$(3 c + 3 \cdot - 2) (c - 3)$

Step 9

Multiply $3$ by $- 2$ .

$(3 c - 6) (c - 3)$

Step 10

Expand $(3 c - 6) (c - 3)$ using the FOIL Method.

Apply the distributive property.

$3 c (c - 3) - 6 (c - 3)$

Apply the distributive property.

$3 c \cdot c + 3 c \cdot - 3 - 6 (c - 3)$

Apply the distributive property.

$3 c \cdot c + 3 c \cdot - 3 - 6 c - 6 \cdot - 3$

Step 11

Simplify and combine like terms.

Simplify each term.

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

$3 (c \cdot c) + 3 c \cdot - 3 - 6 c - 6 \cdot - 3$

Multiply $c$ by $c$ .

$3 c^{2} + 3 c \cdot - 3 - 6 c - 6 \cdot - 3$

Multiply $- 3$ by $3$ .

$3 c^{2} - 9 c - 6 c - 6 \cdot - 3$

Multiply $- 6$ by $- 3$ .

$3 c^{2} - 9 c - 6 c + 18$

Subtract $6 c$ from $- 9 c$ .

$3 c^{2} - 15 c + 18$

Divide ((c^2-4)/(c+3))÷((c+2)/(3(c^2-9)))

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