You provided the expression:

\[
(2x - y + 3)(3x + 2y - 1)
\]

This is a product of two binomials. To expand it, we will use the distributive property (FOIL method). Let's multiply each term in the first binomial by each term in the second binomial.

The steps are as follows:

1. Multiply \( 2x \) by each term in \( (3x + 2y - 1) \):
   \[
   2x(3x) = 6x^2, \quad 2x(2y) = 4xy, \quad 2x(-1) = -2x
   \]

2. Multiply \( -y \) by each term in \( (3x + 2y - 1) \):
   \[
   -y(3x) = -3xy, \quad -y(2y) = -2y^2, \quad -y(-1) = y
   \]

3. Multiply \( 3 \) by each term in \( (3x + 2y - 1) \):
   \[
   3(3x) = 9x, \quad 3(2y) = 6y, \quad 3(-1) = -3
   \]

Now, let's combine all the results:

\[
6x^2 + 4xy - 2x - 3xy - 2y^2 + y + 9x + 6y - 3
\]

Finally, combine like terms:

\[
6x^2 + (4xy - 3xy) + (-2x + 9x) + (-2y^2) + (y + 6y) - 3
\]

Simplify:

\[
6x^2 + xy + 7x - 2y^2 + 7y - 3
\]

So, the expanded form is:

\[
6x^2 + xy + 7x - 2y^2 + 7y - 3
\]

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How can you factor the expanded expression back to its original form?
2. What happens if you change the signs of the terms inside the parentheses?
3. Can you generalize this expansion technique for higher-degree polynomials?
4. How does expanding binomials relate to solving quadratic equations?
5. How do you apply this expansion in real-world problems?

**Tip:** When expanding expressions with multiple variables, group and simplify similar terms carefully to avoid errors.

Expand and simplify the expression (2x−y+3)(3x+2y−1) using the FOIL method. This algebra problem explains binomial expansion, the distributive property, and how to combine like terms. Suitable for students in Grades 8-10, with step-by-step guidance.

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