Sure! Let's solve the given expression step by step. We want to find the square of the expression:
[tex]\[(0, 2x^2 + 5y^2 + 2)\][/tex]
We are specifically interested in squaring only the second part of the tuple:
[tex]\[2x^2 + 5y^2 + 2\][/tex]
Our goal is to compute:
[tex]\[\left(2x^2 + 5y^2 + 2\right)^2\][/tex]
To square the expression, we use the identity [tex]\((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\)[/tex]. Let's apply this step by step.
Here, [tex]\(a = 2x^2\)[/tex], [tex]\(b = 5y^2\)[/tex], and [tex]\(c = 2\)[/tex].
1. Square each term individually:
[tex]\[
(2x^2)^2 = 4x^4
\][/tex]
[tex]\[
(5y^2)^2 = 25y^4
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
2. Compute the cross-product terms:
[tex]\[
2 \cdot (2x^2) \cdot (5y^2) = 20x^2y^2
\][/tex]
[tex]\[
2 \cdot (2x^2) \cdot 2 = 8x^2
\][/tex]
[tex]\[
2 \cdot (5y^2) \cdot 2 = 20y^2
\][/tex]
3. Combine all the terms:
[tex]\[
(2x^2 + 5y^2 + 2)^2 = 4x^4 + 25y^4 + 4 + 20x^2y^2 + 8x^2 + 20y^2
\][/tex]
Thus, the final squared expression is:
[tex]\[\left(2x^2 + 5y^2 + 2\right)^2\][/tex]
Simplified, it remains:
[tex]\[\left(2x^2 + 5y^2 + 2\right)^2\][/tex]
