Let's identify which expression is equivalent to [tex]\( 2x^2 - 2x + 7 \)[/tex] by simplifying each given expression step-by-step.
First Expression:
[tex]\[
(4x + 12) + (2x^2 - 6x + 5)
\][/tex]
Combine like terms:
[tex]\[
= 2x^2 + 4x - 6x + 12 + 5
\][/tex]
[tex]\[
= 2x^2 - 2x + 17
\][/tex]
This simplifies to [tex]\( 2x^2 - 2x + 17 \)[/tex], which is not equivalent to [tex]\( 2x^2 - 2x + 7 \)[/tex].
Second Expression:
[tex]\[
(x^2 - 5x + 13) + (x^2 + 3x - 6)
\][/tex]
Combine like terms:
[tex]\[
= x^2 + x^2 - 5x + 3x + 13 - 6
\][/tex]
[tex]\[
= 2x^2 - 2x + 7
\][/tex]
This simplifies to [tex]\( 2x^2 - 2x + 7 \)[/tex], which is exactly what we are looking for.
Third Expression:
[tex]\[
(4x^2 - 6x + 11) + (2x^2 - 4x + 4)
\][/tex]
Combine like terms:
[tex]\[
= 4x^2 + 2x^2 - 6x - 4x + 11 + 4
\][/tex]
[tex]\[
= 6x^2 - 10x + 15
\][/tex]
This simplifies to [tex]\( 6x^2 - 10x + 15 \)[/tex], which is not equivalent to [tex]\( 2x^2 - 2x + 7 \)[/tex].
Fourth Expression:
[tex]\[
(5x^2 - 8x + 120) + (-3x^2 + 10x - 13)
\][/tex]
Combine like terms:
[tex]\[
= 5x^2 - 3x^2 - 8x + 10x + 120 - 13
\][/tex]
[tex]\[
= 2x^2 + 2x + 107
\][/tex]
This simplifies to [tex]\( 2x^2 + 2x + 107 \)[/tex], which is not equivalent to [tex]\( 2x^2 - 2x + 7 \)[/tex].
Therefore, the expression that is equivalent to [tex]\( 2x^2 - 2x + 7 \)[/tex] is:
[tex]\[
\left( x^2 - 5x + 13 \right) + \left( x^2 + 3x - 6\right)
\][/tex]
So, the correct answer is the second expression.
