Sure! Let’s simplify the given expression step by step: [tex]\((x + 2)^2(x - 4) - x\)[/tex].
1. Expand the first part [tex]\((x + 2)^2 (x - 4)\)[/tex]:
[tex]\((x + 2)^2\)[/tex] can be expanded using the binomial theorem:
[tex]\[
(x + 2)^2 = x^2 + 4x + 4
\][/tex]
2. Multiply this result by [tex]\((x - 4)\)[/tex]:
Now we need to distribute [tex]\((x - 4)\)[/tex] across each term in the polynomial [tex]\(x^2 + 4x + 4\)[/tex]:
[tex]\[
(x^2 + 4x + 4)(x - 4)
\][/tex]
This calculation can be broken into parts:
- Multiplying [tex]\(x^2\)[/tex] by [tex]\((x - 4)\)[/tex]:
[tex]\[
x^2 \cdot (x - 4) = x^3 - 4x^2
\][/tex]
- Multiplying [tex]\(4x\)[/tex] by [tex]\((x - 4)\)[/tex]:
[tex]\[
4x \cdot (x - 4) = 4x^2 - 16x
\][/tex]
- Multiplying [tex]\(4\)[/tex] by [tex]\((x - 4)\)[/tex]:
[tex]\[
4 \cdot (x - 4) = 4x - 16
\][/tex]
Putting all these parts together:
[tex]\[
x^3 - 4x^2 + 4x^2 - 16x + 4x - 16
\][/tex]
Combining like terms:
[tex]\[
x^3 - 16x - 16
\][/tex]
3. Subtract [tex]\(x\)[/tex] from the resulting expression:
[tex]\[
x^3 - 16x - 16 - x
\][/tex]
Simplifying this:
[tex]\[
x^3 - 17x - 16
\][/tex]
Hence, the expression [tex]\((x + 2)^2(x - 4) - x\)[/tex] simplifies to:
[tex]\[
\boxed{x^3 - 17x - 16}
\][/tex]
