Let's perform the polynomial subtraction step-by-step using the expression \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\):
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Distribute the negative sign across the second polynomial: \[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 2x^2 + 6x + 1 - 3x^2 - 3x - 9 \]
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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: \(1 - 9 = -8\)
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Putting it all together, we get: \[ -x^2 + 3x - 8 \]
Now, let's compare this result to the provided examples:
- Example 1: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \) (Incorrect)
- Example 2: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \) (Incorrect)
- Example 3: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \) (Correct)
- Example 4: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \) (Incorrect)
Thus, the correct example demonstrating polynomial subtraction is:
\((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\).