Let's go through the process of matching each pair of polynomials to their sum. We'll pair them and sum them as follows:
1. [tex]\( 12x^2 + 3x + 6 \)[/tex] and [tex]\( -7x^2 - 4x - 2 \)[/tex]:
Adding these together:
[tex]\[
12x^2 + 3x + 6 + (-7x^2 - 4x - 2) = (12x^2 - 7x^2) + (3x - 4x) + (6 - 2) = 5x^2 - x + 4
\][/tex]
2. [tex]\( 2x^2 - x \)[/tex] and [tex]\( -x - 2x^2 - 2 \)[/tex]:
Adding these together:
[tex]\[
2x^2 - x + (-x - 2x^2 - 2) = (2x^2 - 2x^2) + (-x - x) + (-2) = -2x - 2
\][/tex]
3. [tex]\( x + x^2 + 2 \)[/tex] and [tex]\( x^2 - 2 - x \)[/tex]:
Adding these together:
[tex]\[
x + x^2 + 2 + (x^2 - 2 - x) = (x^2 + x^2) + (x - x) + (2 - 2) = 2x^2
\][/tex]
4. [tex]\( x^2 + x \)[/tex] and [tex]\( x^2 + 8x - 2 \)[/tex]:
Adding these together:
[tex]\[
x^2 + x + (x^2 + 8x - 2) = (x^2 + x^2) + (x + 8x) + (-2) = 2x^2 + 9x - 2
\][/tex]
Now, matching each pair of polynomials to their sum:
- [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]
So, the correct pairs are:
[tex]\[
\begin{aligned}
&12x^2 + 3x + 6 \quad \longleftrightarrow \quad 5x^2 - x + 4 \\
&2x^2 - x \quad \longleftrightarrow \quad -2x - 2 \\
& x + x^2 + 2 \quad \longleftrightarrow \quad 2x^2 \\
& x^2 + x \quad \longleftrightarrow \quad 2x^2 + 9x - 2
\end{aligned}
\][/tex]
