Let's begin by simplifying the given polynomial expression:
[tex]\[
\left(5xy^2 + 3x^2 - 7\right) + \left(3x^2y^2 - xy^2 + 3y^2 + 4\right)
\][/tex]
We will combine the like terms from both polynomials one step at a time.
### Step 1: Combine the [tex]\( x^2y^2 \)[/tex] terms
The terms involving [tex]\( x^2y^2 \)[/tex] from both polynomials are:
[tex]\[
5xy^2 \quad \text{and} \quad 3x^2y^2
\][/tex]
These terms sum up to:
[tex]\[
3x^2y^2
\][/tex]
### Step 2: Combine the [tex]\( x^2 \)[/tex] terms
The terms involving [tex]\( x^2 \)[/tex] from both polynomials are:
[tex]\[
3x^2 + 3x^2
\][/tex]
These terms sum up to:
[tex]\[
3x^2
\][/tex]
### Step 3: Combine the [tex]\( xy^2 \)[/tex] terms
The terms involving [tex]\( xy^2 \)[/tex] from both polynomials are:
[tex]\[
5xy^2 - xy^2
\][/tex]
These terms sum up to:
[tex]\[
4xy^2
\][/tex]
### Step 4: Combine the [tex]\( y^2 \)[/tex] terms
The terms involving [tex]\( y^2 \)[/tex] from both polynomials are:
[tex]\[
3y^2
\][/tex]
This term is by itself and doesn't have a counterpart to combine with, so it remains:
[tex]\[
3y^2
\][/tex]
### Step 5: Combine the constant terms
The constant terms from both polynomials are:
[tex]\[
-7 + 4
\][/tex]
These terms sum up to:
[tex]\[
-3
\][/tex]
Now, combining all the simplified parts together, we have the resulting polynomial expression:
[tex]\[
3x^2y^2 + 3x^2 + 4xy^2 + 3y^2 - 3
\][/tex]
Therefore, the polynomial expression:
[tex]\[
\left(5xy^2 + 3x^2 - 7\right) + \left(3x^2y^2 - xy^2 + 3y^2 + 4\right)
\][/tex]
simplifies to:
[tex]\[
3x^2y^2 + 4xy^2 + 3x^2 + 3y^2 - 3
\][/tex]
The correct answer is:
[tex]\[
\boxed{3x^2y^2 + 4xy^2 + 3x^2 + 3y^2 - 3}
\][/tex]
Comparing this with the given options, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
