QuestionDivide as indicated. Simplify the answer. $\begin{array}{l} \frac{a-b}{4 a+4 b} \div \frac{a^{2}-b^{2}}{a^{2}+6 a+9} \\ \frac{a-b}{4 a+4 b} \div \frac{a^{2}-b^{2}}{a^{2}+6 a+9}=\square \end{array}$ $\square$

Studdy Solution

STEP 1

1. The expression involves division of two rational expressions.
2. Simplification will involve factoring polynomials and using properties of division.

STEP 2

1. Rewrite the division as multiplication by the reciprocal.
2. Factor all polynomials in the expression.
3. Simplify the expression by canceling common factors.

STEP 3

Rewrite the division as multiplication by the reciprocal:
$\frac{a-b}{4a+4b} \div \frac{a^2-b^2}{a^2+6a+9} = \frac{a-b}{4a+4b} \times \frac{a^2+6a+9}{a^2-b^2}$

STEP 4

Factor each polynomial:
1. Factor $4a + 4b$ :
$4a + 4b = 4(a + b)$
2. Factor $a^2 - b^2$ (difference of squares):
$a^2 - b^2 = (a - b)(a + b)$
3. Factor $a^2 + 6a + 9$ (perfect square trinomial):
$a^2 + 6a + 9 = (a + 3)^2$

STEP 5

Substitute the factored forms into the expression:
$\frac{a-b}{4(a+b)} \times \frac{(a+3)^2}{(a-b)(a+b)}$
Cancel common factors:
1. Cancel $a-b$ in the numerator and denominator.
2. Cancel $a+b$ in the numerator and denominator.
The simplified expression is:
$\frac{(a+3)^2}{4}$
The simplified answer is:
$\boxed{\frac{(a+3)^2}{4}}$

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