multiply this polynomial into standard form:  -(x-3)(x+1)3(x-2)2

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.

Learn how to multiply and expand the polynomial expression -(x-3)(x+1)3(x-2)2 into its standard form. This detailed step-by-step solution covers essential algebraic concepts such as the distributive property and polynomial expansion. Ideal for students in Grades 9-12.

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