To determine which example correctly demonstrates polynomial subtraction, we need to perform the subtraction \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\) correctly.
Let's perform the calculation step-by-step:
1. Start with the first polynomial:
\[
2x^2 + 6x + 1
\]
2. Subtract the second polynomial:
\[
-(3x^2 + 3x + 9) = -3x^2 - 3x - 9
\]
3. Combine the two results:
\[
(2x^2 + 6x + 1) + (-3x^2 - 3x - 9) = 2x^2 - 3x^2 + 6x - 3x + 1 - 9
\]
4. Simplifying each term:
- \(2x^2 - 3x^2 = -x^2\)
- \(6x - 3x = 3x\)
- \(1 - 9 = -8\)
Putting it all together, we get:
\[
-x^2 + 3x - 8
\]
Now we can check which of the provided responses matches this result:
1. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8\) (This is incorrect.)
2. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) (This is incorrect.)
3. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8\) (This is incorrect.)
4. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (This is correct.)
Therefore, the fourth response correctly demonstrates polynomial subtraction.