Let's go through the expressions step by step:

First Expression:

$(3x^2 + 2x + 4) + (2x^2 - x - 2)$ Combine like terms: $3x^2 + 2x^2 = 5x^2$ $2x - x = x$ $4 - 2 = 2$ Result: $5x^2 + x + 2$

Second Expression:

$(4x + 5y - 6) + (2x + 3y + 4)$ Combine like terms: $4x + 2x = 6x$ $5y + 3y = 8y$ $-6 + 4 = -2$ Result: $6x + 8y - 2$

Third Expression:

$(2x^2 - 3x + 4) - (3x^2 + 4y - 3)$ Distribute the negative sign: $2x^2 - 3x + 4 - 3x^2 - 4y + 3$ Combine like terms: $2x^2 - 3x^2 = -x^2$ $-3x \quad (\text{no like term to combine with})$ $4 - (-4y) = -4y$ $4 + 3 = 7$ Result: $-x^2 - 3x - 4y + 7$

Fourth Expression:

$(5x - 2y - 6) - (3x - 4y - 8)$ Distribute the negative sign: $5x - 2y - 6 - 3x + 4y + 8$ Combine like terms: $5x - 3x = 2x$ $-2y + 4y = 2y$ $-6 + 8 = 2$ Result: $2x + 2y + 2$

Fifth Expression:

$(x + 3y + 6) - (1 - 2x + 2xy - 3)$ Distribute the negative sign: $x + 3y + 6 - 1 + 2x - 2xy + 3$ Combine like terms: $x + 2x = 3x$ $3y \quad (\text{no like term to combine with})$ $-2xy \quad (\text{no like term to combine with})$ $6 - 1 + 3 = 8$ Result: $3x + 3y - 2xy + 8$

Final Answers:

1. $5x^2 + x + 2$
2. $6x + 8y - 2$
3. $-x^2 - 3x - 4y + 7$
4. $2x + 2y + 2$
5. $3x + 3y - 2xy + 8$

Let me know if you'd like further explanations or details on any step.

Related Questions:

1. How do you simplify polynomials by combining like terms?
2. How do you distribute negative signs in algebraic expressions?
3. What are the rules for combining variables with different exponents?
4. How do you approach subtraction of polynomials?
5. How can you check if a polynomial is fully simplified?

Tip:

When simplifying expressions, always start by combining like terms and distributing signs to avoid mistakes in simplification.