Let's simplify the given expression:

$(x - 2y)(y + 3x) - 2xy + 3(x + y - 1)$

Step 1: Expand the first part $(x - 2y)(y + 3x)$

Use the distributive property (FOIL method):

$(x - 2y)(y + 3x) = x(y + 3x) - 2y(y + 3x)$

Now expand each term:

$x(y + 3x) = xy + 3x^2$ $-2y(y + 3x) = -2y^2 - 6xy$

So:

$(x - 2y)(y + 3x) = xy + 3x^2 - 2y^2 - 6xy$

Combine like terms:

$3x^2 - 5xy - 2y^2$

Step 2: Expand the second part $-2xy$

$-2xy$

Step 3: Expand the third part $3(x + y - 1)$

$3(x + y - 1) = 3x + 3y - 3$

Step 4: Combine all the terms

Now put everything together:

$3x^2 - 5xy - 2y^2 - 2xy + 3x + 3y - 3$

Combine the $xy$ terms:

$3x^2 - 7xy - 2y^2 + 3x + 3y - 3$

Final Answer:

$3x^2 - 7xy - 2y^2 + 3x + 3y - 3$

Would you like more details or explanations on any step?

Here are some related questions to explore further:

1. How do you simplify polynomial expressions?
2. What are the steps for expanding binomials using FOIL?
3. How do you combine like terms in algebraic expressions?
4. Can you explain how to distribute negative signs in expressions?
5. What is the role of variables in polynomial equations?

Tip: When expanding expressions, always ensure you combine like terms carefully to avoid errors.