To expand the expression [tex]\((-4x - 5) \left( -x^2 - x + 4 \right)\)[/tex] into a polynomial in standard form, follow these steps:
1. Distribute each term in the first polynomial [tex]\((-4x - 5)\)[/tex] to every term in the second polynomial [tex]\((-x^2 - x + 4)\)[/tex].
2. Distribute [tex]\(-4x\)[/tex]:
- Multiply [tex]\(-4x\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
(-4x) \cdot (-x^2) = 4x^3
\][/tex]
- Multiply [tex]\(-4x\)[/tex] by [tex]\(-x\)[/tex]:
[tex]\[
(-4x) \cdot (-x) = 4x^2
\][/tex]
- Multiply [tex]\(-4x\)[/tex] by [tex]\(4\)[/tex]:
[tex]\[
(-4x) \cdot 4 = -16x
\][/tex]
3. Distribute [tex]\(-5\)[/tex]:
- Multiply [tex]\(-5\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
(-5) \cdot (-x^2) = 5x^2
\][/tex]
- Multiply [tex]\(-5\)[/tex] by [tex]\(-x\)[/tex]:
[tex]\[
(-5) \cdot (-x) = 5x
\][/tex]
- Multiply [tex]\(-5\)[/tex] by [tex]\(4\)[/tex]:
[tex]\[
(-5) \cdot 4 = -20
\][/tex]
4. Combine all the results:
[tex]\[
4x^3 + 4x^2 + (-16x) + 5x^2 + 5x - 20
\][/tex]
5. Combine like terms:
- Combine [tex]\(4x^2\)[/tex] and [tex]\(5x^2\)[/tex]:
[tex]\[
4x^2 + 5x^2 = 9x^2
\][/tex]
- Combine [tex]\(-16x\)[/tex] and [tex]\(5x\)[/tex]:
[tex]\[
-16x + 5x = -11x
\][/tex]
6. The expanded polynomial is:
[tex]\[
4x^3 + 9x^2 - 11x - 20
\][/tex]
Therefore, the expression [tex]\((-4x - 5) \left( -x^2 - x + 4 \right)\)[/tex] expands to the polynomial:
[tex]\[
4x^3 + 9x^2 - 11x - 20
\][/tex]
