Sure! Let's solve the problem step-by-step to find which expression is equivalent to \(2x^2 - 2x + 7\).
We need to check each given option by simplifying and combining their terms to see which one matches the target expression \(2x^2 - 2x + 7\).
1. Option 1:
[tex]\[
(4x + 12) + (2x^2 - 6x + 5)
\][/tex]
Combine the terms:
[tex]\[
4x + 12 + 2x^2 - 6x + 5 = 2x^2 - 2x + 17
\][/tex]
This simplifies to:
[tex]\[
2x^2 - 2x + 17
\][/tex]
This does not match \(2x^2 - 2x + 7\).
2. Option 2:
[tex]\[
(x^2 - 5x + 13) + (x^2 + 3x - 6)
\][/tex]
Combine the terms:
[tex]\[
x^2 - 5x + 13 + x^2 + 3x - 6 = 2x^2 - 2x + 7
\][/tex]
This simplifies to:
[tex]\[
2x^2 - 2x + 7
\][/tex]
This matches \(2x^2 - 2x + 7\).
3. Option 3:
[tex]\[
(4x^2 - 6x + 11) + (2x^2 - 4x + 4)
\][/tex]
Combine the terms:
[tex]\[
4x^2 - 6x + 11 + 2x^2 - 4x + 4 = 6x^2 - 10x + 15
\][/tex]
This simplifies to:
[tex]\[
6x^2 - 10x + 15
\][/tex]
This does not match \(2x^2 - 2x + 7\).
4. Option 4:
[tex]\[
(5x^2 - 8x + 120) + (-3x^2 + 10x - 13)
\][/tex]
Combine the terms:
[tex]\[
5x^2 - 8x + 120 - 3x^2 + 10x - 13 = 2x^2 + 2x + 107
\][/tex]
This simplifies to:
[tex]\[
2x^2 + 2x + 107
\][/tex]
This does not match \(2x^2 - 2x + 7\).
Based on the calculations above, the expression that is equivalent to \(2x^2 - 2x + 7\) is:
[tex]\[
\left( x^2 - 5x + 13 \right) + \left( x^2 + 3x - 6 \right)
\][/tex]
Therefore, the correct answer is Option 2.
