To solve the problem of multiplying [tex]\(-4x\)[/tex] by the polynomial [tex]\(x^3 - x^2 + x - 1\)[/tex], we'll follow a step-by-step multiplication process. Let's distribute [tex]\(-4x\)[/tex] to each term of the polynomial:
### Step-by-Step Multiplication:
1. Multiply [tex]\(-4x\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
-4x \cdot x^3 = -4x^4
\][/tex]
2. Multiply [tex]\(-4x\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
-4x \cdot -x^2 = 4x^3
\][/tex]
3. Multiply [tex]\(-4x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
-4x \cdot x = -4x^2
\][/tex]
4. Multiply [tex]\(-4x\)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[
-4x \cdot -1 = 4x
\][/tex]
Now, let's combine all the terms together to form the resulting polynomial:
[tex]\[
-4x^4 + 4x^3 - 4x^2 + 4x
\][/tex]
So, the correct answer is:
[tex]\[
-4x^4 + 4x^3 - 4x^2 + 4x
\][/tex]
Hence, the result of [tex]\(-4x\)[/tex] multiplied by [tex]\(x^3 - x^2 + x - 1\)[/tex] is [tex]\(-4x^4 + 4x^3 - 4x^2 + 4x\)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{-4x^4 + 4x^3 - 4x^2 + 4x}
\][/tex]
