Answer :
Let's pair each set of polynomials with their sum.
### Pair 1:
First set of polynomials: [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex].
To find the sum, add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ 12x^2 + 3x + 6 + (-7x^2 - 4x - 2) = (12 + -7)x^2 + (3 + -4)x + (6 + -2) \][/tex]
[tex]\[ = 5x^2 - x + 4 \][/tex]
So, the matching sum is [tex]\(5x^2 - x + 4\)[/tex].
### Pair 2:
Second set of polynomials: [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex].
Add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ 2x^2 - x + (-x - 2x^2 - 2) = (2 + -2)x^2 + (-1 + -1)x + 0 + -2 \][/tex]
[tex]\[ = 0x^2 - 2x - 2 \][/tex]
[tex]\[ = -2x - 2 \][/tex]
So, the matching sum is [tex]\(-2x - 2\)[/tex].
### Pair 3:
Third set of polynomials: [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex].
Add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ x + x^2 + 2 + (x^2 - 2 - x) = (1 + 1)x^2 + (1 + -1)x + (2 + -2) \][/tex]
[tex]\[ = 2x^2 + 0x + 0 \][/tex]
[tex]\[ = 2x^2 \][/tex]
So, the matching sum is [tex]\(2x^2\)[/tex].
### Pair 4:
Fourth set of polynomials: [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex].
Add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ x^2 + x + (x^2 + 8x - 2) = (1 + 1)x^2 + (1 + 8)x + 0 + -2 \][/tex]
[tex]\[ = 2x^2 + 9x - 2 \][/tex]
So, the matching sum is [tex]\(2x^2 + 9x - 2\)[/tex].
### Summary of Pairs:
- [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex] => [tex]\(5x^2 - x + 4\)[/tex]
- [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex] => [tex]\(-2x - 2\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] => [tex]\(2x^2\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex] => [tex]\(2x^2 + 9x - 2\)[/tex]
Thus, the correct pairs are as follows:
- [tex]\(12 x^2 + 3 x + 6\)[/tex] and [tex]\(-7 x^2 - 4 x - 2\)[/tex] => [tex]\(5 x^2 - x + 4\)[/tex]
- [tex]\(2 x^2 - x\)[/tex] and [tex]\(-x - 2 x^2 - 2\)[/tex] => [tex]\(-2 x - 2\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] => [tex]\(2 x^2\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8 x - 2\)[/tex] => [tex]\(2 x^2 + 9 x - 2\)[/tex]
### Pair 1:
First set of polynomials: [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex].
To find the sum, add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ 12x^2 + 3x + 6 + (-7x^2 - 4x - 2) = (12 + -7)x^2 + (3 + -4)x + (6 + -2) \][/tex]
[tex]\[ = 5x^2 - x + 4 \][/tex]
So, the matching sum is [tex]\(5x^2 - x + 4\)[/tex].
### Pair 2:
Second set of polynomials: [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex].
Add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ 2x^2 - x + (-x - 2x^2 - 2) = (2 + -2)x^2 + (-1 + -1)x + 0 + -2 \][/tex]
[tex]\[ = 0x^2 - 2x - 2 \][/tex]
[tex]\[ = -2x - 2 \][/tex]
So, the matching sum is [tex]\(-2x - 2\)[/tex].
### Pair 3:
Third set of polynomials: [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex].
Add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ x + x^2 + 2 + (x^2 - 2 - x) = (1 + 1)x^2 + (1 + -1)x + (2 + -2) \][/tex]
[tex]\[ = 2x^2 + 0x + 0 \][/tex]
[tex]\[ = 2x^2 \][/tex]
So, the matching sum is [tex]\(2x^2\)[/tex].
### Pair 4:
Fourth set of polynomials: [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex].
Add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant term.
[tex]\[ x^2 + x + (x^2 + 8x - 2) = (1 + 1)x^2 + (1 + 8)x + 0 + -2 \][/tex]
[tex]\[ = 2x^2 + 9x - 2 \][/tex]
So, the matching sum is [tex]\(2x^2 + 9x - 2\)[/tex].
### Summary of Pairs:
- [tex]\(12x^2 + 3x + 6\)[/tex] and [tex]\(-7x^2 - 4x - 2\)[/tex] => [tex]\(5x^2 - x + 4\)[/tex]
- [tex]\(2x^2 - x\)[/tex] and [tex]\(-x - 2x^2 - 2\)[/tex] => [tex]\(-2x - 2\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] => [tex]\(2x^2\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8x - 2\)[/tex] => [tex]\(2x^2 + 9x - 2\)[/tex]
Thus, the correct pairs are as follows:
- [tex]\(12 x^2 + 3 x + 6\)[/tex] and [tex]\(-7 x^2 - 4 x - 2\)[/tex] => [tex]\(5 x^2 - x + 4\)[/tex]
- [tex]\(2 x^2 - x\)[/tex] and [tex]\(-x - 2 x^2 - 2\)[/tex] => [tex]\(-2 x - 2\)[/tex]
- [tex]\(x + x^2 + 2\)[/tex] and [tex]\(x^2 - 2 - x\)[/tex] => [tex]\(2 x^2\)[/tex]
- [tex]\(x^2 + x\)[/tex] and [tex]\(x^2 + 8 x - 2\)[/tex] => [tex]\(2 x^2 + 9 x - 2\)[/tex]
