Let's solve the expression $(9 - 2x)(x^2 - x)(x)$.

Step 1: Expand the first two factors

We first expand $(9 - 2x)$ and $(x^2 - x)$.

$(9 - 2x)(x^2 - x) = 9(x^2 - x) - 2x(x^2 - x)$

Distribute each term:

$9(x^2 - x) = 9x^2 - 9x$ $-2x(x^2 - x) = -2x^3 + 2x^2$

Now we combine them:

$(9 - 2x)(x^2 - x) = 9x^2 - 9x - 2x^3 + 2x^2$

Step 2: Combine like terms

$9x^2 + 2x^2 - 9x - 2x^3 = 11x^2 - 9x - 2x^3$

So, the result of expanding $(9 - 2x)(x^2 - x)$ is: $-2x^3 + 11x^2 - 9x$

Step 3: Multiply by the third factor, $x$

Now multiply the result by $x$:

$(-2x^3 + 11x^2 - 9x)(x) = -2x^3(x) + 11x^2(x) - 9x(x)$

$= -2x^4 + 11x^3 - 9x^2$

Final Answer:

The fully expanded form of the expression $(9 - 2x)(x^2 - x)(x)$ is:

$-2x^4 + 11x^3 - 9x^2$

Would you like further details on the process or have any questions? Here are some related questions:

1. How do we factor the result $-2x^4 + 11x^3 - 9x^2$?
2. What happens if we substitute $x = 1$ into the expression?
3. Can we apply the distributive property to expand similar expressions with more terms?
4. What is the degree of the polynomial $-2x^4 + 11x^3 - 9x^2$?
5. How would you find the roots of the equation $-2x^4 + 11x^3 - 9x^2 = 0$?

Tip: When multiplying polynomials, always keep track of the powers of $x$ to ensure the terms are combined correctly.