Let's simplify the expression [tex]\( a^4 - (a^2 - 3b^2)^2 \)[/tex] step-by-step.
1. Expand the squared term:
[tex]\[
(a^2 - 3b^2)^2
\][/tex]
To expand this, we use the formula for the square of a binomial, [tex]\((x - y)^2 = x^2 - 2xy + y^2\)[/tex]:
[tex]\[
(a^2 - 3b^2)^2 = (a^2)^2 - 2(a^2)(3b^2) + (3b^2)^2
\][/tex]
Calculating each term:
[tex]\[
(a^2)^2 = a^4
\][/tex]
[tex]\[
2(a^2)(3b^2) = 6a^2b^2
\][/tex]
[tex]\[
(3b^2)^2 = 9b^2
\][/tex]
So,
[tex]\[
(a^2 - 3b^2)^2 = a^4 - 6a^2b^2 + 9b^4
\][/tex]
2. Substitute the expanded term back into the original expression:
[tex]\[
a^4 - (a^2 - 3b^2)^2 = a^4 - \left(a^4 - 6a^2b^2 + 9b^4\right)
\][/tex]
3. Distribute the negative sign:
[tex]\[
a^4 - (a^4 - 6a^2b^2 + 9b^4) = a^4 - a^4 + 6a^2b^2 - 9b^4
\][/tex]
4. Combine like terms:
[tex]\[
a^4 - a^4 + 6a^2b^2 - 9b^4 = 6a^2b^2 - 9b^4
\][/tex]
Therefore, the simplified form of the expression [tex]\( a^4 - (a^2 - 3b^2)^2 \)[/tex] is:
[tex]\[
6a^2b^2 - 9b^4
\][/tex]
