Let's simplify the given expression:

\[
-3x(-2x^2y + 4xy - 5y^2) + 2y(-x^3 + x^2 + 6y)
\]

We'll simplify it step by step:

### Step 1: Distribute \(-3x\) across the first polynomial
\[
-3x(-2x^2y + 4xy - 5y^2) = (-3x)(-2x^2y) + (-3x)(4xy) + (-3x)(-5y^2)
\]
Simplify each term:
- \((-3x)(-2x^2y) = 6x^3y\)
- \((-3x)(4xy) = -12x^2y\)
- \((-3x)(-5y^2) = 15xy^2\)

So, the expression becomes:
\[
6x^3y - 12x^2y + 15xy^2
\]

### Step 2: Distribute \(2y\) across the second polynomial
\[
2y(-x^3 + x^2 + 6y) = (2y)(-x^3) + (2y)(x^2) + (2y)(6y)
\]
Simplify each term:
- \((2y)(-x^3) = -2x^3y\)
- \((2y)(x^2) = 2x^2y\)
- \((2y)(6y) = 12y^2\)

So, the expression becomes:
\[
-2x^3y + 2x^2y + 12y^2
\]

### Step 3: Combine the two simplified expressions
Now, add the results from Steps 1 and 2 together:
\[
(6x^3y - 12x^2y + 15xy^2) + (-2x^3y + 2x^2y + 12y^2)
\]
Group like terms together:
- \(6x^3y - 2x^3y = 4x^3y\)
- \(-12x^2y + 2x^2y = -10x^2y\)
- \(15xy^2 + 12y^2 = 15xy^2 + 12y^2\) (no like terms here)

So the final simplified expression is:
\[
4x^3y - 10x^2y + 15xy^2 + 12y^2
\]

This is the fully simplified form of the given expression.

Would you like further details or have any questions?

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Here are some related questions you might find helpful:

1. How do you distribute a term across a polynomial?
2. What are like terms in algebraic expressions?
3. How do you factor out common factors in algebraic expressions?
4. How can you simplify complex expressions with multiple variables?
5. What are the steps to combining like terms in polynomials?

**Tip:** When simplifying expressions, always double-check your signs (positive or negative) after distribution to avoid common mistakes.

Learn how to simplify the expression -3x(-2x²y + 4xy - 5y²) + 2y(-x³ + x² + 6y) step-by-step. This problem involves algebraic manipulation of polynomials, distribution, and combining like terms. Suitable for high school students.

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