To expand the expression [tex]\(\left(3 a^2 b^2 + 5 x^2 y^2\right)^2\)[/tex], we can use the binomial theorem, which states that [tex]\((u + v)^2 = u^2 + 2uv + v^2\)[/tex].
Here, [tex]\(u = 3 a^2 b^2\)[/tex] and [tex]\(v = 5 x^2 y^2\)[/tex].
Let's expand the expression step-by-step:
1. Calculate [tex]\(u^2\)[/tex]:
[tex]\[
u^2 = \left(3 a^2 b^2\right)^2 = \left(3\right)^2 \left(a^2\right)^2 \left(b^2\right)^2 = 9 a^4 b^4
\][/tex]
2. Calculate [tex]\(2uv\)[/tex]:
[tex]\[
2uv = 2 \times (3 a^2 b^2) \times (5 x^2 y^2) = 2 \times 3 \times 5 \times a^2 b^2 \times x^2 y^2 = 30 a^2 b^2 x^2 y^2
\][/tex]
3. Calculate [tex]\(v^2\)[/tex]:
[tex]\[
v^2 = \left(5 x^2 y^2\right)^2 = \left(5\right)^2 \left(x^2\right)^2 \left(y^2\right)^2 = 25 x^4 y^4
\][/tex]
Now, combine these results to form the expanded expression:
[tex]\[
(u + v)^2 = u^2 + 2uv + v^2
\][/tex]
Substitute the expressions [tex]\(u^2\)[/tex], [tex]\(2uv\)[/tex], and [tex]\(v^2\)[/tex] back into the formula:
[tex]\[
(3 a^2 b^2 + 5 x^2 y^2)^2 = 9 a^4 b^4 + 30 a^2 b^2 x^2 y^2 + 25 x^4 y^4
\][/tex]
So, the expanded form of [tex]\(\left(3 a^2 b^2 + 5 x^2 y^2\right)^2\)[/tex] is:
[tex]\[
\boxed{9 a^4 b^4 + 30 a^2 b^2 x^2 y^2 + 25 x^4 y^4}
\][/tex]
