Certainly! Let's perform the indicated operation step-by-step.
Given expression:
[tex]\[
\frac{a^2 - 4c^2}{a + 2c} \div (a + 2c) \cdot \frac{3c}{a - 2c}
\][/tex]
Step 1: Factor the numerator of the first fraction.
The expression \(a^2 - 4c^2\) is a difference of squares which can be factored as:
[tex]\[
a^2 - 4c^2 = (a + 2c)(a - 2c)
\][/tex]
Step 2: Substitute the factored form into the expression.
[tex]\[
\frac{(a + 2c)(a - 2c)}{a + 2c} \div (a + 2c) \cdot \frac{3c}{a - 2c}
\][/tex]
Step 3: Simplify the first fraction by canceling out the common terms.
[tex]\[
\frac{(a + 2c)(a - 2c)}{a + 2c} = a - 2c \quad (\text{since \(a + 2c\) cancels out})
\][/tex]
The expression now becomes:
[tex]\[
(a - 2c) \div (a + 2c) \cdot \frac{3c}{a - 2c}
\][/tex]
Step 4: Rewrite the division as multiplication by the reciprocal.
[tex]\[
(a - 2c) \cdot \frac{1}{a + 2c} \cdot \frac{3c}{a - 2c}
\][/tex]
Step 5: Simplify the expression by canceling out \(a - 2c\) in the numerator and the denominator.
[tex]\[
= \frac{(a - 2c) \cdot 3c}{(a + 2c)(a - 2c)}
\][/tex]
[tex]\[
= \frac{3c}{a + 2c}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{3c}{a + 2c}
\][/tex]
