To determine which expression is equivalent to the given polynomial expression:
[tex]\[
(5xy^2 + 3x^2 - 7) + (3x^2y^2 - xy^2 + 3y^2 + 4),
\][/tex]
we will combine like terms step by step.
1. Collect terms involving [tex]\(x^2 y^2\)[/tex]:
- From the second polynomial: [tex]\(3x^2 y^2\)[/tex]
- Sum: [tex]\(3x^2 y^2\)[/tex]
2. Collect terms involving [tex]\(xy^2\)[/tex]:
- From the first polynomial: [tex]\(5xy^2\)[/tex]
- From the second polynomial: [tex]\(-xy^2\)[/tex]
- Sum: [tex]\(5xy^2 - xy^2 = 4xy^2\)[/tex]
3. Collect terms involving [tex]\(y^2\)[/tex]:
- From the second polynomial: [tex]\(3y^2\)[/tex]
- Sum: [tex]\(3y^2\)[/tex]
4. Collect terms involving [tex]\(x^2\)[/tex]:
- From the first polynomial: [tex]\(3x^2\)[/tex]
- Sum: [tex]\(3x^2\)[/tex]
5. Collect constants:
- From the first polynomial: [tex]\(-7\)[/tex]
- From the second polynomial: [tex]\(4\)[/tex]
- Sum: [tex]\(-7 + 4 = -3\)[/tex]
By combining all these, the equivalent polynomial expression is:
[tex]\[
3x^2y^2 + 4xy^2 + 3y^2 + 3x^2 - 3
\][/tex]
Thus, the correct answer is:
C. [tex]\(3 x^2 y^2 + 4 x y^2 + 3 x^2 + 3 y^2 - 3\)[/tex]
