Let's go through each of the given polynomial additions step-by-step to determine their equivalent expressions:
1. First, consider the expression [tex]\((3x^2 - 7x + 14) + (5x^2 + 4x - 6)\)[/tex]. When we combine like terms, we get:
- Coefficient for [tex]\(x^2\)[/tex]: [tex]\(3 + 5 = 8\)[/tex]
- Coefficient for [tex]\(x\)[/tex]: [tex]\(-7 + 4 = -3\)[/tex]
- Constant term: [tex]\(14 - 6 = 8\)[/tex]
Therefore, the resulting expression is [tex]\(8x^2 - 3x + 8\)[/tex].
Looking at the table:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline A & B & C \\
\hline$-8 x^2-3 x+4$ & $8 x^2-3 x+8$ & $8 x^2+3 x-4$ \\
\hline
\end{tabular}
\][/tex]
This expression matches with [tex]\(B\)[/tex].
2. Next, consider the expression [tex]\((2x^2 - 5x - 3) + (-10x^2 + 2x + 7)\)[/tex]. When we combine like terms, we get:
- Coefficient for [tex]\(x^2\)[/tex]: [tex]\(2 - 10 = -8\)[/tex]
- Coefficient for [tex]\(x\)[/tex]: [tex]\(-5 + 2 = -3\)[/tex]
- Constant term: [tex]\(-3 + 7 = 4\)[/tex]
Therefore, the resulting expression is [tex]\(-8x^2 - 3x + 4\)[/tex].
Looking at the table again, this expression matches with [tex]\(A\)[/tex].
3. Finally, consider the expression [tex]\((12x^2 - 2x - 13) + (-4x^2 + 5x + 9)\)[/tex]. When we combine like terms, we get:
- Coefficient for [tex]\(x^2\)[/tex]: [tex]\(12 - 4 = 8\)[/tex]
- Coefficient for [tex]\(x\)[/tex]: [tex]\(-2 + 5 = 3\)[/tex]
- Constant term: [tex]\(-13 + 9 = -4\)[/tex]
Therefore, the resulting expression is [tex]\(8x^2 + 3x - 4\)[/tex].
Looking once again at the table, this expression matches with [tex]\(C\)[/tex].
To sum up, we have:
- [tex]\((3 x^2 - 7 x + 14) + (5 x^2 + 4 x - 6)\)[/tex] is equivalent to expression [tex]\(B\)[/tex]
- [tex]\((2 x^2 - 5 x - 3) + (-10 x^2 + 2 x + 7)\)[/tex] is equivalent to expression [tex]\(A\)[/tex]
- [tex]\((12 x^2 - 2 x - 13) + (-4 x^2 + 5 x + 9)\)[/tex] is equivalent to expression [tex]\(C\)[/tex]
Therefore, the correct answers to fill in the boxes are:
[tex]\[
\left(3 x^2 - 7 x + 14\right) + \left(5 x^2 + 4 x - 6\right) \text{ is equivalent to expression } \boxed{B}
\][/tex]
[tex]\[
\left(2 x^2 - 5 x - 3\right) + \left(-10 x^2 + 2 x + 7\right) \text{ is equivalent to expression } \boxed{A}
\][/tex]
[tex]\[
\left(12 x^2 - 2 x - 13\right) + \left(-4 x^2 + 5 x + 9\right) \text{ is equivalent to expression } \boxed{C}
\][/tex]
