Let's take each pair of polynomials one by one and find their sums step-by-step.
### 1. Adding \(12x^2 + 3x + 6\) and \(-7x^2 - 4x - 2\)
#### Step-by-Step Addition:
- Combine the coefficients of \(x^2\):
\(12x^2 + (-7x^2) = (12 - 7)x^2 = 5x^2\)
- Combine the coefficients of \(x\):
\(3x + (-4x) = (3 - 4)x = -x\)
- Combine the constants:
\(6 + (-2) = 6 - 2 = 4\)
Thus, the sum of the first pair of polynomials is:
[tex]\[5x^2 - x + 4\][/tex]
### 2. Adding \(2x^2 - x\) and \(-x - 2x^2 - 2\)
#### Step-by-Step Addition:
- Combine the coefficients of \(x^2\):
\(2x^2 + (-2x^2) = (2 - 2)x^2 = 0\)
- Combine the coefficients of \(x\):
\(-x + (-x) = (-1 - 1)x = -2x\)
- Combine the constants:
[tex]\[0 - 2 = -2\][/tex]
Thus, the sum of the second pair of polynomials is:
[tex]\[-2x - 2\][/tex]
### 3. Adding \(x^3 + x^2 + 2\) and \(x^2 - 2 - x^3\)
#### Step-by-Step Addition:
- Combine the coefficients of \(x^3\):
\(x^3 + (-x^3) = (1 - 1)x^3 = 0\)
- Combine the coefficients of \(x^2\):
\(x^2 + x^2 = (1 + 1)x^2 = 2x^2\)
- Combine the constants:
\(2 + (-2) = 2 - 2 = 0\)
Thus, the sum of the third pair of polynomials is:
[tex]\[2x^2\][/tex]
### 4. Adding \(x^2 + x\) and \(x^2 + 8x - 2\)
#### Step-by-Step Addition:
- Combine the coefficients of \(x^2\):
\(x^2 + x^2 = (1 + 1)x^2 = 2x^2\)
- Combine the coefficients of \(x\):
\(x + 8x = (1 + 8)x = 9x\)
- Combine the constants:
\(0 + (-2) = -2\)
Thus, the sum of the fourth pair of polynomials is:
[tex]\[2x^2 + 9x - 2\][/tex]
### Final Results:
We have found the sums of the pairs of polynomials as follows:
1. \(5x^2 - x + 4\)
2. \(-2x - 2\)
3. \(2x^2\)
4. [tex]\(2x^2 + 9x - 2\)[/tex]
