The given expression is:

$xy - 2xy + 3x^2y - 4xy^2 + 2xy^2$

Step 1: Combine Like Terms

We can group the terms that are similar:

$xy - 2xy = -xy$ $-4xy^2 + 2xy^2 = -2xy^2$

Thus, the simplified polynomial becomes:

$-xy + 3x^2y - 2xy^2$

Step 2: Substitute $x = 4$ and $y = -2$

Now we substitute the values $x = 4$ and $y = -2$ into the simplified expression:

$-xy = -(4)(-2) = 8$ $3x^2y = 3(4^2)(-2) = 3(16)(-2) = -96$ $-2xy^2 = -2(4)(-2^2) = -2(4)(4) = -32$

Step 3: Add the Results

Now, sum up the terms:

$8 + (-96) + (-32) = 8 - 96 - 32 = -120$

So the correct answer is: $\boxed{-xy - 2xy^2 + 3x^2y; -120}$

The correct option matches the second choice.

Would you like more details or have any further questions?

Here are 5 related questions for practice:

1. How do you identify like terms in a polynomial?
2. What happens when you substitute negative values into polynomial expressions?
3. How do exponents affect the calculation when substituting values into a polynomial?
4. How can you quickly simplify polynomials with multiple like terms?
5. How does factoring polynomials relate to combining like terms?

Tip: Always simplify expressions before substituting values to avoid unnecessary complications.