To multiply the polynomial $- (x - 3)(x + 1)3(x - 2)2$ into standard form, follow these steps:

Step 1: Distribute the factors

Let's first expand the expression $(x + 1)(x - 2)^2$.

The square of $(x - 2)$ is: $(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4$

Now multiply this by $(x + 1)$: $(x + 1)(x^2 - 4x + 4) = x(x^2 - 4x + 4) + 1(x^2 - 4x + 4)$ $= x^3 - 4x^2 + 4x + x^2 - 4x + 4$ $= x^3 - 3x^2 + 4$

Step 2: Multiply by the remaining factors

Next, we multiply this result by $(x - 3)$: $(x - 3)(x^3 - 3x^2 + 4) = x(x^3 - 3x^2 + 4) - 3(x^3 - 3x^2 + 4)$ $= x^4 - 3x^3 + 4x - 3x^3 + 9x^2 - 12$ $= x^4 - 6x^3 + 9x^2 + 4x - 12$

Step 3: Apply the negative sign

Finally, multiply the entire expression by $-1$ to account for the leading negative sign: $- (x^4 - 6x^3 + 9x^2 + 4x - 12) = -x^4 + 6x^3 - 9x^2 - 4x + 12$

Final Answer

The polynomial in standard form is: $-x^4 + 6x^3 - 9x^2 - 4x + 12$

Would you like more details or have any questions about this solution?

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Here are five related questions you might find interesting:

1. How do you factor a polynomial like $x^4 - 6x^3 + 9x^2 + 4x - 12$?
2. What are the roots of the polynomial $x^4 - 6x^3 + 9x^2 + 4x - 12$?
3. How would the solution change if the sign on one of the factors were different?
4. How can we determine the degree of a polynomial?
5. What are some methods for expanding binomials quickly?

Tip: When expanding polynomials, it's often helpful to first handle any squared or higher-power terms before dealing with linear terms.