To determine which example correctly demonstrates polynomial subtraction, we need to perform the subtraction \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) \).
1. Rewrite the subtraction:
\[
2x^2 + 6x + 1 - (3x^2 + 3x + 9)
\]
This is the same as:
\[
2x^2 + 6x + 1 - 3x^2 - 3x - 9
\]
2. Combine like terms:
- For \( x^2 \):
\[
2x^2 - 3x^2 = -1x^2 \text{ or } -x^2
\]
- For \( x \):
\[
6x - 3x = 3x
\]
- For the constant term:
\[
1 - 9 = -8
\]
Putting it all together, we get:
\[
-x^2 + 3x - 8
\]
Now, let's evaluate the given responses:
1. **Response 1**: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \) (Incorrect)
2. **Response 2**: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \) (Incorrect)
3. **Response 3**: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \) (Correct)
4. **Response 4**: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \) (Incorrect)
Therefore, the correct response is:
**Response 3: (2x² + 6x + 1) − (3x² + 3x + 9) = −x² + 3x − 8**.