Question

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

Answer 1

To expand the expression \((−3x−1)(2x^2−x−2)\), we will use the distributive property (also known as the FOIL method for binomials, but we’ll apply it to a trinomial).

1. Begin by distributing \(-3x\) to each term in the second polynomial \(2x^2 − x − 2\):
\[
-3x \cdot 2x^2 = -6x^3
\]
\[
-3x \cdot (-x) = 3x^2
\]
\[
-3x \cdot (-2) = 6x
\]

So, combining these results gives:
\[
-6x^3 + 3x^2 + 6x
\]

2. Next, distribute \(-1\) to each term in the polynomial \(2x^2 − x − 2\):
\[
-1 \cdot 2x^2 = -2x^2
\]
\[
-1 \cdot (-x) = x
\]
\[
-1 \cdot (-2) = 2
\]

So, combining these results gives:
\[
-2x^2 + x + 2
\]

3. Now, we can combine all the results from both distributions:
\[
-6x^3 + (3x^2 - 2x^2) + (6x + x) + 2
\]

4. Simplify by combining like terms:
\[
-6x^3 + (3x^2 - 2x^2) = -6x^3 + 1x^2
\]
\[
(6x + x) = 7x
\]

Hence, we get the combined polynomial:
\[
-6x^3 + x^2 + 7x + 2
\]

5. Finally, write the polynomial in standard form:
\[
-6x^3 + x^2 + 7x + 2
\]

Thus, the expanded expression in standard form is:
\[
\boxed{-6x^3 + x^2 + 7x + 2}
\]