To match each pair of polynomials to their sum, let's analyze each provided polynomial pair and their corresponding sums.
1. Pair: [tex]\(12 x^2+3 x+6\)[/tex] and [tex]\(-7 x^2-4 x-2\)[/tex]
Solution:
[tex]\[ (12 x^2 + 3x + 6) + (-7 x^2 - 4x - 2) \][/tex]
[tex]\[ = (12 x^2 - 7 x^2) + (3x - 4x) + (6 - 2) \][/tex]
[tex]\[ = 5 x^2 - x + 4 \][/tex]
So, the sum is [tex]\(5 x^2 - x + 4\)[/tex].
2. Pair: [tex]\(2 x^2 - x\)[/tex] and [tex]\(-x - 2 x^2 - 2\)[/tex]
Solution:
[tex]\[ (2 x^2 - x) + (-2 x^2 - x - 2) \][/tex]
[tex]\[ = (2 x^2 - 2 x^2) + (-x - x) + (-2) \][/tex]
[tex]\[ = 0x^2 - 2x - 2 \][/tex]
[tex]\[ = -2 x - 2 \][/tex]
So, the sum is [tex]\(-2 x - 2\)[/tex].
3. Pair: [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex]
Solution:
[tex]\[ (x + x^2 + 2) + (x^2 - 2 - x) \][/tex]
[tex]\[ = (x^2 + x^2) + (x - x) + (2 - 2) \][/tex]
[tex]\[ = 2 x^2 \][/tex]
So, the sum is [tex]\(2 x^2\)[/tex].
4. Pair: [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8 x - 2\)[/tex]
Solution:
[tex]\[ (x^2 + x) + (x^2 + 8x - 2) \][/tex]
[tex]\[ = (x^2 + x^2) + (x + 8x) + (-2) \][/tex]
[tex]\[ = 2 x^2 + 9x - 2 \][/tex]
So, the sum is [tex]\(2 x^2 + 9 x - 2\)[/tex].
Putting it all together:
- [tex]\(12 x^2+3 x+6\)[/tex] and [tex]\(-7 x^2-4 x-2\)[/tex] sum to [tex]\(\boxed{5 x^2 - x + 4}\)[/tex]
- [tex]\(2 x^2 - x\)[/tex] and [tex]\(-x - 2 x^2 - 2\)[/tex] sum to [tex]\(\boxed{-2 x - 2}\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] sum to [tex]\(\boxed{2 x^2}\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8 x - 2\)[/tex] sum to [tex]\(\boxed{2 x^2 + 9 x - 2}\)[/tex]
