Question

Expand the expression to a polynomial in standard form:
left parenthesis, 4, x, minus, 1, right parenthesis, left parenthesis, minus, 3, x, squared, minus, 2, x, minus, 6, right parenthesis
(4x−1)(−3x
2
−2x−6)

Answer 1

To expand the expression \((4x - 1)(-3x^2 - 2x - 6)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s how it works step by step:

1. **Distribute \(4x\) over \(-3x^2 - 2x - 6\)**:
\[
4x(-3x^2) + 4x(-2x) + 4x(-6) = -12x^3 - 8x^2 - 24x
\]

2. **Distribute \(-1\) over \(-3x^2 - 2x - 6\)**:
\[
-1(-3x^2) + (-1)(-2x) + (-1)(-6) = 3x^2 + 2x + 6
\]

3. **Combine the results from step 1 and step 2**:
\[
-12x^3 - 8x^2 - 24x + 3x^2 + 2x + 6
\]

4. **Combine like terms**:
- For \(x^3\) terms: \(-12x^3\)
- For \(x^2\) terms: \(-8x^2 + 3x^2 = -5x^2\)
- For \(x\) terms: \(-24x + 2x = -22x\)
- For constant terms: \(6\)

Thus, the expanded polynomial in standard form is:
\[
-12x^3 - 5x^2 - 22x + 6
\]