To check which example correctly demonstrates polynomial subtraction, we need to perform the subtraction on the given polynomials:
\[ (2x^{2}+6x+1) - (3x^{2}+3x+9). \]
We can distribute the negative sign into the second polynomial:
\[ = 2x^2 + 6x + 1 - 3x^2 - 3x - 9. \]
Now, we can group the like terms together:
\[ = (2x^2 - 3x^2) + (6x - 3x) + (1 - 9). \]
Calculating each group:
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For the \(x^2\) terms: \[ 2x^2 - 3x^2 = -1x^2 \text{ or } -x^2. \]
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For the \(x\) terms: \[ 6x - 3x = 3x. \]
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For the constant terms: \[ 1 - 9 = -8. \]
Putting it all together, we have:
\[ -x^2 + 3x - 8. \]
Now we compare this result with the provided answer choices:
- \((-x^2 + 3x - 8)\)
- \((-5x^2 + 3x - 8)\)
- \( (x^2 + 3x + 8) \)
- \( (5x^2 + 9x + 10) \)
The first option matches our result exactly. Therefore, the correct example that demonstrates polynomial subtraction is:
\[ \boxed{(2x^{2}+6x+1)-(3x^{2}+3x+9)=-x^{2}+3x-8}. \]