Simplify each of the following expressions.
Your answer should be a polynomial, that is, a sum of terms which are each a coeficient times powers of the variables.

(-4x^2 + 6xy - 3y^2) * (8x^2 - 8xy) 

To simplify the given expression \((-4x^2 + 6xy - 3y^2)(8x^2 - 8xy)\), we will apply the distributive property (i.e., distribute each term from the first polynomial to each term from the second polynomial). 

### Step-by-step expansion:
The original expression is:
\[
(-4x^2 + 6xy - 3y^2) \cdot (8x^2 - 8xy)
\]

We'll multiply each term from the first polynomial by each term in the second polynomial:

1. \(-4x^2 \cdot 8x^2 = -32x^4\)
2. \(-4x^2 \cdot (-8xy) = 32x^3y\)
3. \(6xy \cdot 8x^2 = 48x^3y\)
4. \(6xy \cdot (-8xy) = -48x^2y^2\)
5. \(-3y^2 \cdot 8x^2 = -24x^2y^2\)
6. \(-3y^2 \cdot (-8xy) = 24xy^3\)

### Combine like terms:
Now, let's combine the terms:
- There are no other \(x^4\) terms, so \(-32x^4\) stays as it is.
- The \(x^3y\) terms are \(32x^3y + 48x^3y = 80x^3y\).
- The \(x^2y^2\) terms are \(-48x^2y^2 - 24x^2y^2 = -72x^2y^2\).
- The \(xy^3\) term is \(24xy^3\).

### Final simplified expression:
\[
-32x^4 + 80x^3y - 72x^2y^2 + 24xy^3
\]

This is the simplified polynomial.

Would you like further clarification, or have any questions? 

Here are 5 related questions for practice:
1. Simplify \((3x^2 - 2xy)(2x - y)\).
2. Simplify \((x^2 + y^2)(x + y)\).
3. Multiply \((2x - y)(x + 3y)\) and simplify.
4. Expand and simplify \((x^2 - 3xy + y^2)(x + y)\).
5. Multiply \((a^2 + ab)(b^2 - ab)\).

**Tip:** When multiplying polynomials, always distribute each term in the first polynomial to each term in the second polynomial, and combine like terms carefully.

Simplify each of the following expressions. Your answer should be a polynomial, that is, a sum of terms which are each a coefficient times powers of the variables. (-4x^2 + 6xy - 3y^2) * (8x^2 - 8xy)

Learn how to simplify the polynomial expression (-4x^2 + 6xy - 3y^2)(8x^2 - 8xy) with a step-by-step breakdown. This problem explores the distributive property and combines like terms, suitable for high school students in Grades 9-12.

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