To find the sum of the given polynomials and group like terms together, we'll follow these steps:
1. Write down the given polynomials:
[tex]\[
10 x^2 y + 2 x y^2 - 4 x^2 - 4 x^2 y
\][/tex]
2. Combine like terms:
- First, notice that \(10 x^2 y \) and \(-4 x^2 y \) are like terms because they both contain \( x^2 y \).
- Next, \(2 x y^2\) has no other like term.
- Finally, \(-4 x^2 \) has no other like term.
3. Combine the like terms:
- Combine \(10 x^2 y\) and \(-4 x^2 y\):
[tex]\[
10 x^2 y - 4 x^2 y = 6 x^2 y
\][/tex]
- The other terms \(2 x y^2\) and \(-4 x^2\) remain unchanged.
Therefore, the grouped polynomial is:
[tex]\[
6 x^2 y + 2 x y^2 - 4 x^2
\][/tex]
Now, let's match this simplified polynomial with the provided choices:
1. [tex]\[
\left[\left(-4 x^2\right)+\left(-4 x^2 y\right)+10 x^2 y\right]+2 x y^2
\][/tex]
2. [tex]\[
10 x^2 y + 2 x y^2 + \left[\left(-4 x^2\right)+\left(-4 x^2 y\right)\right]
\][/tex]
3. [tex]\[
\left(-4 x^2\right)+2 x y^2+\left[10 x^2 y+\left(-4 x^2 y\right)\right]
\][/tex]
4. [tex]\[
\left[10 x^2 y+2 x y^2+\left(-4 x^2 y\right)\right]+\left(-4 x^2\right]
\][/tex]
5. [tex]\[
\left[10 x^2 y + 2 x y^2 + \left(-4 x^2 y\right)\right] + \left(-4 x^2\right)
\][/tex]
From the choices, the correct expression that matches our simplified polynomial:
[tex]\[
6 x^2 y + 2 x y^2 - 4 x^2
\][/tex]
is option 5.
Thus, the answer is:
[tex]\[
\left[10 x^2 y + 2 x y^2 + \left(-4 x^2 y \right) \right ] + \left(-4 x^2 \right)
\][/tex]
This indicates the sum of the polynomials with like terms grouped together.
