Certainly! Let's expand the given expression [tex]\((2x + 1)(-2x^2 - x - 1)\)[/tex] step by step.
### Step 1: Distribute [tex]\(2x\)[/tex] to each term in the second polynomial
First, we need to distribute [tex]\(2x\)[/tex] across the terms in [tex]\(-2x^2 - x - 1\)[/tex]:
[tex]\[
2x \cdot (-2x^2) + 2x \cdot (-x) + 2x \cdot (-1)
\][/tex]
So,
[tex]\[
2x \cdot (-2x^2) = -4x^3
\][/tex]
[tex]\[
2x \cdot (-x) = -2x^2
\][/tex]
[tex]\[
2x \cdot (-1) = -2x
\][/tex]
Combining these, we get:
[tex]\[
-4x^3 - 2x^2 - 2x
\][/tex]
### Step 2: Distribute [tex]\(1\)[/tex] to each term in the second polynomial
Next, we distribute [tex]\(1\)[/tex] across the terms in [tex]\(-2x^2 - x - 1\)[/tex]:
[tex]\[
1 \cdot (-2x^2) + 1 \cdot (-x) + 1 \cdot (-1)
\][/tex]
So,
[tex]\[
1 \cdot (-2x^2) = -2x^2
\][/tex]
[tex]\[
1 \cdot (-x) = -x
\][/tex]
[tex]\[
1 \cdot (-1) = -1
\][/tex]
Combining these, we get:
[tex]\[
-2x^2 - x - 1
\][/tex]
### Step 3: Combine all terms
Now we need to add all the results together:
[tex]\[
-4x^3 - 2x^2 - 2x + (-2x^2 - x - 1)
\][/tex]
Combine the like terms:
[tex]\[
= -4x^3 + (-2x^2 - 2x^2) + (-2x - x) - 1
\][/tex]
[tex]\[
= -4x^3 - 4x^2 - 3x - 1
\][/tex]
### Final Answer
The expanded expression in standard polynomial form is:
[tex]\[
-4x^3 - 4x^2 - 3x - 1
\][/tex]
