To determine which example correctly demonstrates polynomial addition, we need to add the like terms from the two polynomials.
Given the expression:
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9)\)
We can add the coefficients of each like term:
1. For \(x^2\) terms:
\(2x^2 + 3x^2 = 5x^2\)
2. For \(x\) terms:
\(6x + 3x = 9x\)
3. For constant terms:
\(1 + 9 = 10\)
So, the sum is:
\(5x^2 + 9x + 10\)
Now, let's analyze the responses:
1. **First response:**
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) - **Correct!**
2. **Second response:**
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10\) - **Incorrect!** (The \(x^2\) terms were added incorrectly.)
3. **Third response:**
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10\) - **Incorrect!** (The \(x^2\) terms were added incorrectly, and there's an extraneous \(x^4\) term.)
4. **Fourth response:**
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8\) - **Incorrect!** (This expression does not properly reflect the addition of the original polynomials.)
Thus, the correct answer is the **first response**:
\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\).