Let's work through each part of the problem by expanding the expressions and combining like terms.

Solution:

1. a) $2(2x + 3y) + 3(3x - 2y)$
   Expanding each part: $2 \cdot 2x + 2 \cdot 3y + 3 \cdot 3x - 3 \cdot 2y = 4x + 6y + 9x - 6y$ Combine like terms: $(4x + 9x) + (6y - 6y) = 13x$ Answer: $13x$

2. b) $-5(x - 2y) + 2(2x - 5y)$
   Expanding each part: $-5 \cdot x + (-5) \cdot (-2y) + 2 \cdot 2x + 2 \cdot (-5y) = -5x + 10y + 4x - 10y$ Combine like terms: $(-5x + 4x) + (10y - 10y) = -x$ Answer: $-x$

3. c) $a(2a + b) - b(b - 2a)$
   Expanding each part: $a \cdot 2a + a \cdot b - b \cdot b + b \cdot 2a = 2a^2 + ab - b^2 + 2ab$ Combine like terms: $2a^2 + (ab + 2ab) - b^2 = 2a^2 + 3ab - b^2$ Answer: $2a^2 + 3ab - b^2$

4. d) $-b(2b + a) - a(a - 2b)$
   Expanding each part: $-b \cdot 2b - b \cdot a - a \cdot a + a \cdot 2b = -2b^2 - ab - a^2 + 2ab$ Combine like terms: $-2b^2 + (-ab + 2ab) - a^2 = -2b^2 + ab - a^2$ Answer: $-2b^2 + ab - a^2$

5. e) $3x(2y - x) + 2y(3x - 2y)$
   Expanding each part: $3x \cdot 2y - 3x \cdot x + 2y \cdot 3x - 2y \cdot 2y = 6xy - 3x^2 + 6xy - 4y^2$ Combine like terms: $-3x^2 + (6xy + 6xy) - 4y^2 = -3x^2 + 12xy - 4y^2$ Answer: $-3x^2 + 12xy - 4y^2$

6. f) $-4y(x + y) - 2y(3x - 2y)$
   Expanding each part: $-4y \cdot x - 4y \cdot y - 2y \cdot 3x + 2y \cdot 2y = -4xy - 4y^2 - 6xy + 4y^2$ Combine like terms: $(-4xy - 6xy) + (-4y^2 + 4y^2) = -10xy$ Answer: $-10xy$

Let me know if you would like further explanations for any part.

Related Questions

1. How do you handle combining terms with different variables in expressions?
2. What techniques can be used to simplify polynomials more efficiently?
3. Can you explain the distributive property used in polynomial expansion?
4. How would the process change if we included terms with higher powers, like $x^3$ or $y^3$?
5. What are common mistakes to watch out for when simplifying polynomial expressions?

Tip:

Always double-check signs when distributing terms, as a single error with signs can change the entire outcome of the simplification.