To expand the expression \((x^2 + 2x - 4)(-3x^2 - 4x - 4)\), we will apply the distributive property (also known as the FOIL method for binomials, though in this case, we have polynomials to multiply).
We will multiply each term in the first polynomial by each term in the second polynomial.
Let's expand step by step:
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Multiply \(x^2\) by each term in \(-3x^2 - 4x - 4\): \[ x^2 \cdot (-3x^2) = -3x^4 \] \[ x^2 \cdot (-4x) = -4x^3 \] \[ x^2 \cdot (-4) = -4x^2 \]
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Multiply \(2x\) by each term in \(-3x^2 - 4x - 4\): \[ 2x \cdot (-3x^2) = -6x^3 \] \[ 2x \cdot (-4x) = -8x^2 \] \[ 2x \cdot (-4) = -8x \]
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Multiply \(-4\) by each term in \(-3x^2 - 4x - 4\): \[ -4 \cdot (-3x^2) = 12x^2 \] \[ -4 \cdot (-4x) = 16x \] \[ -4 \cdot (-4) = 16 \]
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Combine all the terms together: \[ -3x^4 + (-4x^3) + (-6x^3) + (-4x^2) + (-8x^2) + (-8x) + (12x^2) + (16x) + 16 \]
Now, let's combine like terms:
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For \(x^4\): \(-3x^4\)
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For \(x^3\): \(-4x^3 - 6x^3 = -10x^3\)
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For \(x^2\): \(-4x^2 - 8x^2 + 12x^2 = 0x^2\)
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For \(x\): \(-8x + 16x = 8x\)
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Constant term: \(16\)
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Write the final polynomial in standard form: \[ -3x^4 - 10x^3 + 8x + 16 \]
Thus, the expanded expression in standard form is:
\[ \boxed{-3x^4 - 10x^3 + 8x + 16} \]