To expand the expression \((-4x - 3)(3x^2 - x - 1)\), we will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.
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Distributing \(-4x\):
- \(-4x \cdot 3x^2 = -12x^3\)
- \(-4x \cdot (-x) = 4x^2\)
- \(-4x \cdot (-1) = 4x\)
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Distributing \(-3\):
- \(-3 \cdot 3x^2 = -9x^2\)
- \(-3 \cdot (-x) = 3x\)
- \(-3 \cdot (-1) = 3\)
Now, combine all the resulting products:
\[ -12x^3 + 4x^2 + 4x - 9x^2 + 3x + 3 \]
Next, combine like terms:
- The \(x^2\) terms: \(4x^2 - 9x^2 = -5x^2\)
- The \(x\) terms: \(4x + 3x = 7x\)
Putting it all together, we have:
\[ -12x^3 - 5x^2 + 7x + 3 \]
Thus, the polynomial in standard form is:
\[ \boxed{-12x^3 - 5x^2 + 7x + 3} \]
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