Let's carefully analyze Marcus's polynomial addition step by step to determine the error he made.
Given expressions:
1. \( 3x^2 - 2y^2 + 5x \)
2. \( 4x^2 + 12y^2 - 7x \)
To find the combined expression, we should add the corresponding terms from each polynomial.
1. Combine the \( x^2 \) terms:
[tex]\[
(3x^2) + (4x^2) = 7x^2
\][/tex]
2. Combine the \( y^2 \) terms:
[tex]\[
(-2y^2) + (12y^2) = 10y^2
\][/tex]
3. Combine the \( x \) terms:
[tex]\[
5x + (-7x) = -2x
\][/tex]
Thus, the correctly combined expression should be:
[tex]\[
7x^2 + 10y^2 - 2x
\][/tex]
However, Marcus stated that:
[tex]\[
(3x^2 - 2y^2 + 5x) + (4x^2 + 12y^2 - 7x) = 7x^2 - 10y^2 - 2x
\][/tex]
Comparing the correct combined expression \( 7x^2 + 10y^2 - 2x \) with Marcus's result \( 7x^2 - 10y^2 - 2x \), we can see that Marcus combined the \(-2y^2\) and \(12y^2\) terms incorrectly. He ended up with \(-10y^2\) instead of the correct \(10y^2\).
Therefore, the error Marcus made is:
He combined the terms [tex]\(-2 y^2\)[/tex] and [tex]\(12 y^2\)[/tex] incorrectly.
