To find which expression each of the following differences of polynomial expressions is equivalent to, we simplify each step-by-step.
### Simplification Steps:
1. Simplify [tex]\((3x^2 - 6x + 11) - (10x^2 - 4x + 6)\)[/tex]:
[tex]\[
\begin{align*}
=& (3x^2 - 6x + 11) - 10x^2 + 4x - 6 \\
=& (3x^2 - 10x^2) + (-6x + 4x) + (11 - 6) \\
=& -7x^2 - 2x + 5 \\
\end{align*}
\][/tex]
This is equivalent to expression [tex]\(A\)[/tex].
2. Simplify [tex]\((-3x^2 - 5x - 3) - (-10x^2 - 7x + 2)\)[/tex]:
[tex]\[
\begin{align*}
=& (-3x^2 - 5x - 3) + 10x^2 + 7x - 2 \\
=& (-3x^2 + 10x^2) + (-5x + 7x) + (-3 - 2) \\
=& 7x^2 + 2x - 5 \\
\end{align*}
\][/tex]
This is equivalent to expression [tex]\(C\)[/tex].
3. Simplify [tex]\((12x^2 + 6x - 5) - (5x^2 + 8x - 12)\)[/tex]:
[tex]\[
\begin{align*}
=& (12x^2 + 6x - 5) - 5x^2 - 8x + 12 \\
=& (12x^2 - 5x^2) + (6x - 8x) + (-5 + 12) \\
=& 7x^2 - 2x + 7 \\
\end{align*}
\][/tex]
This is equivalent to expression [tex]\(B\)[/tex].
### Final Answers:
[tex]\[
\begin{array}{l}
(3x^2 - 6x + 11) - (10x^2 - 4x + 6) \text{ is equivalent to expression } A \\
(-3x^2 - 5x - 3) - (-10x^2 - 7x + 2) \text{ is equivalent to expression } C \\
(12x^2 + 6x - 5) - (5x^2 + 8x - 12) \text{ is equivalent to expression } B \\
\end{array}
\][/tex]
So, the completed statements are:
[tex]\[
\left(3x^2 - 6x + 11\right) - \left(10x^2 - 4x + 6\right) \text{ is equivalent to expression} \ A
\][/tex]
[tex]\[
\left(-3x^2 - 5x - 3\right) - \left(-10x^2 - 7x + 2\right) \text{ is equivalent to expression} \ C
\][/tex]
[tex]\[
\left(12x^2 + 6x - 5\right) - \left(5x^2 + 8x - 12\right) \text{ is equivalent to expression} \ B
\][/tex]
