To determine which expression is equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex], we start by understanding that division of fractions is equivalent to multiplying by the reciprocal of the second fraction.
Given expression:
[tex]\[
\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}
\][/tex]
First rewrite the division as a multiplication by the reciprocal:
[tex]\[
\frac{c^2-4}{c+3} \times \frac{3(c^2-9)}{c+2}
\][/tex]
Next, we'll simplify the components of the expression.
1. Factorize [tex]\(c^2 - 4\)[/tex] and [tex]\(c^2 - 9\)[/tex] using the difference of squares:
[tex]\[
c^2 - 4 = (c + 2)(c - 2)
\][/tex]
[tex]\[
c^2 - 9 = (c + 3)(c - 3)
\][/tex]
2. Substitute the factored forms into the expression:
[tex]\[
\frac{(c + 2)(c - 2)}{c + 3} \times \frac{3((c + 3)(c - 3))}{c + 2}
\][/tex]
3. Simplify the expression by canceling out the common terms:
[tex]\[
= \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3(c + 3)(c - 3)}{c + 2}
\][/tex]
Notice that [tex]\(c + 2\)[/tex] in the numerator and denominator can be canceled:
[tex]\[
= \frac{(c - 2)}{c + 3} \times 3(c + 3)(c - 3)
\][/tex]
Now cancel [tex]\(c + 3\)[/tex] in the numerator and denominator:
[tex]\[
= \frac{c - 2}{1} \times 3(c - 3)
\][/tex]
This simplifies further to:
[tex]\[
= 3(c - 2) / (c - 3)
\][/tex]
Thus, the simplified expression is:
[tex]\[
3 \cdot \frac{(c - 2)}{(c - 3)}
\][/tex]
So, the expression equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex] is:
[tex]\[
\boxed{\frac{c^2-4}{c+3} \cdot \frac{3 \left( c^2-9 \right)}{c+2}}
\][/tex]
Rewrite this expression for clarity as:
[tex]\[
3 \cdot \frac{(c-2)}{(c-3)}
\][/tex]
