Which example correctly demonstrates polynomial subtraction? (2x2+6x+1)−(3x2+3x+9)=x2+3x+8

To solve the polynomial subtraction \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\), we will start by distributing the negative sign across the second polynomial:

\[
(2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 2x^2 + 6x + 1 - 3x^2 - 3x - 9
\]

Now we can group like terms:

1. For the \(x^2\) terms:
\(2x^2 - 3x^2 = -x^2\)

2. For the \(x\) terms:
\(6x - 3x = 3x\)

3. For the constant terms:
\(1 - 9 = -8\)

Putting it all together, we get:

\[
-x^2 + 3x - 8
\]

Now, let's compare this result with the options provided:

1. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \quad \text{(incorrect)}\)
2. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \quad \text{(incorrect)}\)
3. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \quad \text{(correct)}\)
4. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \quad \text{(incorrect)}\)

Thus, the correct demonstration of polynomial subtraction is:

\(\boxed{-x^2 + 3x - 8}\) (which corresponds to option 3).