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:
-
For the \(x^2\) terms: \(2x^2 - 3x^2 = -x^2\)
-
For the \(x\) terms: \(6x - 3x = 3x\)
-
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:
- \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \quad \text{(incorrect)}\)
- \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \quad \text{(incorrect)}\)
- \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \quad \text{(correct)}\)
- \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \quad \text{(incorrect)}\)
Thus, the correct demonstration of polynomial subtraction is:
\(\boxed{-x^2 + 3x - 8}\) (which corresponds to option 3).