To solve the given polynomial expression, we need to combine like terms step-by-step.
Given polynomial expression:
[tex]\[
\left(5 x y^2 + 3 x^2 - 7\right) + \left(3 x^2 y^2 - x y^2 + 3 y^2 + 4\right)
\][/tex]
First, distribute any constants and group the like terms together. We'll identify and combine terms involving [tex]\(x^2y^2\)[/tex], [tex]\(xy^2\)[/tex], [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex], and the constant terms.
Step-by-step:
1. Group the [tex]\(x^2y^2\)[/tex] terms:
[tex]\[
(3 x^2 y^2)
\][/tex]
There is only one term with [tex]\(x^2y^2\)[/tex].
2. Group the [tex]\(xy^2\)[/tex] terms:
[tex]\[
(5 x y^2) - (x y^2) = 4 x y^2
\][/tex]
Combining these, we have [tex]\(4 x y^2\)[/tex].
3. Group the [tex]\(x^2\)[/tex] terms:
[tex]\[
3 x^2
\][/tex]
Again, there is only one term with [tex]\(x^2\)[/tex].
4. Group the [tex]\(y^2\)[/tex] terms:
[tex]\[
(3 y^2)
\][/tex]
There is only one term with [tex]\(y^2\)[/tex].
5. Combine the constant terms:
[tex]\[
(-7) + 4 = -3
\][/tex]
Putting all these grouped terms together, the simplified expression is:
[tex]\[
3 x^2 y^2 + 4 x y^2 + 3 x^2 + 3 y^2 - 3
\][/tex]
Thus, the expression that is equivalent to the given polynomial is:
C. [tex]\(3 x^2 y^2 + 4 x y^2 + 3 x^2 + 3 y^2 - 3\)[/tex]
