To expand the expression [tex]\((-4x - 3)(3x^2 - x - 1)\)[/tex] into a polynomial in standard form, you'll want to distribute each term in the first polynomial to each term in the second polynomial. Here is the detailed step-by-step process:
1. Distribute [tex]\(-4x\)[/tex] to each term in [tex]\(3x^2 - x - 1\)[/tex]:
- [tex]\( -4x \cdot 3x^2 = -12x^3 \)[/tex]
- [tex]\( -4x \cdot (-x) = 4x^2 \)[/tex]
- [tex]\( -4x \cdot (-1) = 4x \)[/tex]
2. Distribute [tex]\(-3\)[/tex] to each term in [tex]\(3x^2 - x - 1\)[/tex]:
- [tex]\( -3 \cdot 3x^2 = -9x^2 \)[/tex]
- [tex]\( -3 \cdot (-x) = 3x \)[/tex]
- [tex]\( -3 \cdot (-1) = 3 \)[/tex]
Now you combine all these results:
[tex]\[
(-4x - 3)(3x^2 - x - 1) = -12x^3 + 4x^2 + 4x - 9x^2 + 3x + 3
\][/tex]
3. Combine like terms:
- For [tex]\(x^3\)[/tex] term: [tex]\(-12x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 9x^2 = -5x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(4x + 3x = 7x\)[/tex]
- Constant term: [tex]\(3\)[/tex]
So, the expanded expression in standard form is:
[tex]\[
-12x^3 - 5x^2 + 7x + 3
\][/tex]
Therefore, the polynomial in standard form is:
[tex]\[
-12x^3 - 5x^2 + 7x + 3
\][/tex]
