To find the sum of the polynomials \(10x^2y + 2xy^2 - 4x^2\) and \(-4x^2y\), we should group and combine the like terms from each polynomial step by step. Let's work through this process:
1. Write down the given polynomials:
[tex]\[
P_1 = 10x^2y + 2xy^2 - 4x^2
\][/tex]
[tex]\[
P_2 = -4x^2y
\][/tex]
2. Group the like terms:
Notice that in \(P_1\) and \(P_2\), we have terms involving \(x^2y\), \(xy^2\), and \(x^2\).
3. Combine the like terms:
- The terms involving \(x^2y\):
[tex]\[
10x^2y + (-4x^2y) = (10 - 4)x^2y = 6x^2y
\][/tex]
- The terms involving \(xy^2\):
[tex]\[
2xy^2
\][/tex]
(since there's only one term involving \(xy^2\), it remains unchanged)
- The terms involving \(x^2\):
[tex]\[
-4x^2
\][/tex]
(again, since it’s the only term involving \(x^2\), it remains the same)
4. Write the sum of the polynomials by combining all the terms:
[tex]\[
6x^2y + 2xy^2 - 4x^2
\][/tex]
Therefore, the expression that shows the sum of the polynomials with like terms grouped together is:
[tex]\[ 6x^2y + 2xy^2 - 4x^2 \][/tex]
