To determine which expression has the like terms grouped together, let's carefully examine each given option:
1. Option 1: [tex]\( 10 x^2 y + 2 x y^2 - 4 x^2 - 4 x^2 y \)[/tex]
- This is the original expression without any grouping of like terms.
2. Option 2: [tex]\( \left[(-4 x^2) + (-4 x^2 y) + 10 x^2 y\right] + 2 x y^2 \)[/tex]
- Grouping within the brackets shows an attempt to combine [tex]\((-4 x^2)\)[/tex], [tex]\((-4 x^2 y)\)[/tex], and [tex]\(10 x^2 y\)[/tex]. However, the terms inside and outside the brackets aren't consistently grouped with like terms.
3. Option 3: [tex]\( 10 x^2 y + 2 x y^2 + \left[(-4 x^2) + (-4 x^2 y)\right] \)[/tex]
- Similar to Option 2, this expression has partial grouping inside the brackets but the overall combination still needs further simplification.
4. Option 4: [tex]\( (-4 x^2) + 2 x y^2 + \left[10 x^2 y + (-4 x^2 y)\right] \)[/tex]
- Inside the brackets, we see [tex]\(10 x^2 y\)[/tex] combined with [tex]\(-4 x^2 y\)[/tex]. Outside the brackets, [tex]\(-4 x^2\)[/tex] and [tex]\(2 x y^2\)[/tex] are presented, ensuring that all like terms are clearly grouped.
5. Option 5: [tex]\( \left[10 x^2 y + 2 x y^2 + (-4 x^2 y)\right] + (-4 x^2) \)[/tex]
- Mainly groups terms inside the brackets, yet the placement is less clear than in Option 4.
Analyzing carefully:
- Option 4 is the best choice as it groups like terms logically. Inside the bracket, [tex]\(10 x^2 y - 4 x^2 y\)[/tex] are polynomial-like terms combined together. Similarly, outside the bracket are other terms which are not affected by [tex]\(x y\)[/tex] or [tex]\(x^2\)[/tex].
Thus, the expression that shows the sum of the polynomials with like terms grouped together is:
[tex]\[ \boxed{\left(-4 x^2\right) + 2 x y^2 + \left[10 x^2 y + \left(-4 x^2 y\right)\right]} \][/tex]
