Let's simplify the expression step by step:

\[
(x-1)x(x^2+x+1) + x(x+2)(2-x)
\]

### Step 1: Expand each term separately.

#### First term: \((x-1)x(x^2 + x + 1)\)
Distribute \(x\) to the polynomial \((x^2 + x + 1)\):
\[
x(x^2 + x + 1) = x^3 + x^2 + x
\]

Now distribute \((x-1)\) to \((x^3 + x^2 + x)\):
\[
(x-1)(x^3 + x^2 + x) = x(x^3 + x^2 + x) - 1(x^3 + x^2 + x)
\]
\[
= x^4 + x^3 + x^2 - x^3 - x^2 - x
\]
Simplify:
\[
= x^4 - x
\]

#### Second term: \(x(x+2)(2-x)\)
Expand \((x+2)(2-x)\) using the distributive property:
\[
(x+2)(2-x) = x(2-x) + 2(2-x) = 2x - x^2 + 4 - 2x
\]
Simplify:
\[
= -x^2 + 4
\]

Now distribute \(x\):
\[
x(-x^2 + 4) = -x^3 + 4x
\]

### Step 2: Combine both terms.

We now have:
\[
x^4 - x + (-x^3 + 4x)
\]
Simplify by combining like terms:
\[
x^4 - x^3 - x + 4x = x^4 - x^3 + 3x
\]

### Final simplified expression:
\[
x^4 - x^3 + 3x
\]

Would you like a deeper explanation of any step? Here are 5 related questions:

1. How do you expand polynomials with multiple terms?
2. What is the distributive property and how does it apply here?
3. Can you show more examples of simplifying polynomial expressions?
4. How do you factor polynomial expressions?
5. What are the key rules for combining like terms?

**Tip**: When expanding terms, always be careful to apply the distributive property step by step to avoid mistakes.

The query submitted by the user:(x-1)x(x^2+x+1)+x(x+2)(2-x)

Explore how to simplify the polynomial expression (x-1)x(x^2+x+1)+x(x+2)(2-x) using the distributive property and combining like terms. A step-by-step guide for students in Grades 9-12.

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