To determine which expression is equivalent to \(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\), we need to perform the division operation by transforming it into a multiplication by the reciprocal.
1. Rewrite the division as a multiplication:
[tex]\[
\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)} \quad \Rightarrow \quad \frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}
\][/tex]
2. Factorize the polynomials where possible:
- \(c^2-4\) is a difference of squares: \(c^2 - 4 = (c - 2)(c + 2)\)
- \(c^2 - 9\) is also a difference of squares: \(c^2 - 9 = (c - 3)(c + 3)\)
Thus, the expression becomes:
[tex]\[
\frac{(c-2)(c+2)}{c+3} \cdot \frac{3(c-3)(c+3)}{c+2}
\][/tex]
3. Simplify the expression by canceling common factors:
[tex]\[
\frac{(c-2)(c+2)}{c+3} \cdot \frac{3(c-3)(c+3)}{c+2} = \frac{(c-2)\cancel{(c+2)}}{\cancel{c+3}} \cdot \frac{3(c-3)\cancel{(c+3)}}{\cancel{c+2}}
\][/tex]
After canceling the common factors \((c+2)\) and \((c+3)\), we are left with:
[tex]\[
(c-2) \cdot 3(c-3) = 3(c-2)(c-3)
\][/tex]
4. Conclude the simplified expression:
[tex]\[
3(c-2)(c-3)
\][/tex]
Now, we need to see which option matches our simplified form. We previously transformed the division into a multiplication:
Comparing with the given options:
1. \(\frac{c+3}{c^2-4} \div \frac{c+2}{3(c^2-9)}\) is incorrect.
2. \(\frac{c^2-4}{c+3} \div \frac{3(c^2-9)}{c+2}\) is incorrect.
3. \(\frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}\) is correct (this matches our step).
4. \(\frac{c+3}{c^2-4} \cdot \frac{c+2}{3(c^2-9)}\) is incorrect.
Thus, the correct equivalent expression is:
[tex]\(\frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}\)[/tex].
