Let's solve the given question step by step to find the sum of each pair of polynomials.
1. Pair: [tex]\( 12x^2 + 3x + 6 \)[/tex] and [tex]\( -7x^2 - 4x - 2 \)[/tex]
- To find their sum, we add the corresponding coefficients of [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and the constant term:
[tex]\[
(12x^2 + 3x + 6) + (-7x^2 - 4x - 2) = (12x^2 - 7x^2) + (3x - 4x) + (6 - 2)
\][/tex]
Simplifying, we get:
[tex]\[
5x^2 - x + 4
\][/tex]
2. Pair: [tex]\( 2x^2 - x \)[/tex] and [tex]\( -x - 2x^2 - 2 \)[/tex]
- Add the corresponding coefficients:
[tex]\[
(2x^2 - x) + (-x - 2x^2 - 2) = (2x^2 - 2x^2) + (-x - x) + (-2)
\][/tex]
Simplifying, we get:
[tex]\[
0x^2 - 2x - 2 = -2x - 2
\][/tex]
3. Pair: [tex]\( x + x^2 + 2 \)[/tex] and [tex]\( x^2 - 2 - x \)[/tex]
- Add the corresponding coefficients:
[tex]\[
(x + x^2 + 2) + (x^2 - 2 - x) = (x^2 + x^2) + (x - x) + (2 - 2)
\][/tex]
Simplifying, we get:
[tex]\[
2x^2 + 0x + 0 = 2x^2
\][/tex]
4. Pair: [tex]\( x^2 + x \)[/tex] and [tex]\( x^2 + 8x - 2 \)[/tex]
- Add the corresponding coefficients:
[tex]\[
(x^2 + x) + (x^2 + 8x - 2) = (x^2 + x^2) + (x + 8x) + (-2)
\][/tex]
Simplifying, we get:
[tex]\[
2x^2 + 9x - 2
\][/tex]
In summary, matching each pair of polynomials to their sum, we get:
- [tex]\( 12x^2 + 3x + 6 \)[/tex] and [tex]\( -7x^2 - 4x - 2 \)[/tex] sum to [tex]\( 5x^2 - x + 4 \)[/tex]
- [tex]\( 2x^2 - x \)[/tex] and [tex]\( -x - 2x^2 - 2 \)[/tex] sum to [tex]\( -2x - 2 \)[/tex]
- [tex]\( x + x^2 + 2 \)[/tex] and [tex]\( x^2 - 2 - x \)[/tex] sum to [tex]\( 2x^2 \)[/tex]
- [tex]\( x^2 + x \)[/tex] and [tex]\( x^2 + 8x - 2 \)[/tex] sum to [tex]\( 2x^2 + 9x - 2 \)[/tex]
Thus, the matched answers are correctly formatted as:
- [tex]\( 5x^2 - x + 4 \)[/tex]
- [tex]\( -2x - 2 \)[/tex]
- [tex]\( 2x^2 \)[/tex]
- [tex]\( 2x^2 + 9x - 2 \)[/tex]
