Let's simplify the given expression step by step.
First, let's write down the original expression:
[tex]\[
(5x^3 y^2 - 3xy + 2) + (2x^3 y^2 - 3x^2 y^2 + 4xy - 7)
\][/tex]
Next, we will combine the like terms. Like terms are terms that have exactly the same variables raised to the same powers.
1. Combining [tex]\(x^3 y^2\)[/tex] terms:
[tex]\[
5x^3 y^2 + 2x^3 y^2 = 7x^3 y^2
\][/tex]
2. Combining [tex]\(x^2 y^2\)[/tex] terms:
[tex]\[
0x^2 y^2 + (-3x^2 y^2) = -3x^2 y^2
\][/tex]
Note: The first polynomial does not have an [tex]\(x^2 y^2\)[/tex] term, so we consider it as [tex]\(0 \cdot x^2 y^2\)[/tex].
3. Combining [tex]\(xy\)[/tex] terms:
[tex]\[
-3xy + 4xy = xy
\][/tex]
4. Combining constant terms:
[tex]\[
2 + (-7) = -5
\][/tex]
Putting all these together, the simplified expression is:
[tex]\[
7x^3 y^2 - 3x^2 y^2 + xy - 5
\][/tex]
Given the options:
1. [tex]\(3 x^3 y^2 + 3 x^2 y^2 - 7 xy + 9\)[/tex]
2. [tex]\(7 x^3 y^2 - 3 x^2 y^2 + x y - 5\)[/tex]
3. [tex]\(7 x^3 y^2 - 6 x^2 y^2 + 4 x y - 5\)[/tex]
4. [tex]\(7 x^6 y^4 - 5\)[/tex]
The simplified expression matches with:
[tex]\[
7 x^3 y^2 - 3 x^2 y^2 + x y - 5
\][/tex]
So the correct answer is:
[tex]\[
7 x^3 y^2 - 3 x^2 y^2 + x y - 5
\][/tex]
