Answer :
Answer:
To factorize the expression \( 9a^2 - (a^2 - 4)^2 \), let's proceed step by step:
1. **Recognize the structure:**
- The expression \( (a^2 - 4)^2 \) can be expanded using the identity \( (a^2 - b^2)^2 = (a^2 - b^2)(a^2 - b^2) \).
- Here, \( a^2 = a^2 \) and \( b^2 = 2 \), so \( (a^2 - 2)^2 = a^4 - 4a^2 + 4 \).
2. **Substitute and simplify:**
- Substitute \( (a^2 - 4)^2 = a^4 - 8a^2 + 16 \) into the original expression:
\[
9a^2 - (a^4 - 8a^2 + 16)
\]
- Distribute the negative sign:
\[
9a^2 - a^4 + 8a^2 - 16
\]
3. **Combine like terms:**
- Combine the terms involving \( a^2 \):
\[
9a^2 + 8a^2 - a^4 - 16 = 17a^2 - a^4 - 16
\]
4. **Factorize the resulting expression:**
- Factor out a common factor, if possible. Here, notice that \( a^4 - 16 \) is a difference of squares:
\[
a^4 - 16 = (a^2 - 4)(a^2 + 4)
\]
- So the expression \( 17a^2 - a^4 - 16 \) can be written as:
\[
17a^2 - (a^4 - 16) = 17a^2 - (a^2 - 4)(a^2 + 4)
\]
Therefore, the factorized form of \( 9a^2 - (a^2 - 4)^2 \) is \( \boxed{(3a - (a^2 - 4))(3a + (a^2 - 4))} \).