Certainly! Let's expand the expression [tex]\(\left(a^2 x + b y^2\right)^2\)[/tex] step-by-step.
### Step-by-Step Solution
1. Start with the given expression:
[tex]\[
\left(a^2 x + b y^2\right)^2
\][/tex]
2. Apply the square of a binomial formula:
Recall the square of a binomial formula [tex]\((u + v)^2 = u^2 + 2uv + v^2\)[/tex].
Let's set [tex]\(u = a^2 x\)[/tex] and [tex]\(v = b y^2\)[/tex].
3. Calculate each term:
- The first term is the square of [tex]\(u\)[/tex]:
[tex]\[
u^2 = \left(a^2 x\right)^2 = a^4 x^2
\][/tex]
- The second term is twice the product of [tex]\(u\)[/tex] and [tex]\(v\)[/tex]:
[tex]\[
2uv = 2 \left(a^2 x\right) \left(b y^2\right) = 2 a^2 x \cdot b y^2 = 2 a^2 b x y^2
\][/tex]
- The third term is the square of [tex]\(v\)[/tex]:
[tex]\[
v^2 = \left(b y^2\right)^2 = b^2 y^4
\][/tex]
4. Combine all the terms:
Bring all the terms together to form the expanded expression:
[tex]\[
\left(a^2 x + b y^2\right)^2 = a^4 x^2 + 2 a^2 b x y^2 + b^2 y^4
\][/tex]
Therefore, the expanded form of the expression [tex]\(\left(a^2 x + b y^2\right)^2\)[/tex] is:
[tex]\[
a^4 x^2 + 2 a^2 b x y^2 + b^2 y^4
\][/tex]
This concludes the step-by-step expansion process.
