To find the correct demonstration of polynomial subtraction, we need to perform the subtraction step-by-step on the first expression provided:
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9)
\]
Step 1: Distribute the negative sign to the second polynomial:
\[
2x^2 + 6x + 1 - 3x^2 - 3x - 9
\]
Step 2: Combine like terms:
- Combine \(2x^2\) and \(-3x^2\):
\[
2x^2 - 3x^2 = -x^2
\]
- Combine \(6x\) and \(-3x\):
\[
6x - 3x = 3x
\]
- Combine \(1\) and \(-9\):
\[
1 - 9 = -8
\]
Putting it all together, we get:
\[
-x^2 + 3x - 8
\]
Next, let's compare this result with the provided options:
1. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) (Incorrect)
2. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8\) (Incorrect)
3. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (Correct)
4. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8\) (Incorrect)
The correct example that demonstrates polynomial subtraction is:
\((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (Correct option 3).