To expand the expression \((-x + 4)(3x^2 - 2x - 7)\), we can use the distributive property (also known as the FOIL method for binomials). We will multiply each term in the first binomial by each term in the second polynomial.
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Multiply \(-x\) by each term in the second polynomial:
- \(-x \cdot 3x^2 = -3x^3\)
- \(-x \cdot (-2x) = 2x^2\)
- \(-x \cdot (-7) = 7x\)
So, multiplying \(-x\) gives us: \[ -3x^3 + 2x^2 + 7x \]
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Multiply \(4\) by each term in the second polynomial:
- \(4 \cdot 3x^2 = 12x^2\)
- \(4 \cdot (-2x) = -8x\)
- \(4 \cdot (-7) = -28\)
So, multiplying \(4\) gives us: \[ 12x^2 - 8x - 28 \]
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Combine all the results: Combine the results from the two steps:
\[ (-3x^3 + 2x^2 + 7x) + (12x^2 - 8x - 28) \]
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Combine like terms:
- The \(x^3\) term: \(-3x^3\)
- The \(x^2\) terms: \(2x^2 + 12x^2 = 14x^2\)
- The \(x\) terms: \(7x - 8x = -x\)
- The constant term: \(-28\)
Putting this all together, we have:
\[ -3x^3 + 14x^2 - x - 28 \]
Thus, the expression \((-x + 4)(3x^2 - 2x - 7)\) in standard polynomial form is:
\[ \boxed{-3x^3 + 14x^2 - x - 28} \]