Which example correctly demonstrates polynomial subtraction?

Let's perform the polynomial subtraction step-by-step using the expression \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\):

1. Distribute the negative sign across the second polynomial:
\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 2x^2 + 6x + 1 - 3x^2 - 3x - 9
\]

2. Combine like terms:
- For \(x^2\) terms: \(2x^2 - 3x^2 = -1x^2\) (or \(-x^2\))
- For \(x\) terms: \(6x - 3x = 3x\)
- For constant terms: \(1 - 9 = -8\)

3. Putting it all together, we get:
\[
-x^2 + 3x - 8
\]

Now, let's compare this result to the provided examples:

1. **Example 1:** \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \) (Incorrect)
2. **Example 2:** \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \) (Incorrect)
3. **Example 3:** \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \) (Correct)
4. **Example 4:** \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \) (Incorrect)

Thus, the correct example demonstrating polynomial subtraction is:

\((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\).