Let's simplify the given expression step by step:

$(-4a^2 + 3ab + 6b^2) + (-2a^2 + 2ab -5b^2)$

Step 1: Remove Parentheses

Since we are adding, we can simply remove the parentheses:

$-4a^2 + 3ab + 6b^2 - 2a^2 + 2ab - 5b^2$

Step 2: Group Like Terms

• $a^2$ terms: $-4a^2 - 2a^2 = -6a^2$
• $ab$ terms: $3ab + 2ab = 5ab$
• $b^2$ terms: $6b^2 - 5b^2 = 1b^2$

Step 3: Write the Final Expression

$-6a^2 + 5ab + b^2$

Thus, the simplified expression is:
$\mathbf{-6a^2 + 5ab + b^2}$

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Would you like a detailed explanation or any clarifications?

Here are some related questions to explore further:

1. How do you simplify expressions with more than three variables?
2. How do you factorize quadratic expressions like this one?
3. What are some common mistakes when simplifying algebraic expressions?
4. How do you handle subtraction of polynomials instead of addition?
5. Can we represent this expression geometrically?

Tip:

Always double-check your work by verifying like terms before combining them!