https://file.aiquickdraw.com/mathbot/file/1729463254156gs35bg3c.jpg

Let's break down the polynomial addition problem from the image:

The expression given is:
\[
(xy - 9x + z^2y) + (7x - x^2 - xy)
\]

### Step 1: Group like terms
Group similar terms (terms that contain the same variables):
- \(xy\) and \(-xy\)
- \(-9x\) and \(7x\)
- \(z^2y\) (no like terms here)
- \(-x^2\) (no like terms here)

### Step 2: Combine the like terms
Now, we combine them:
- \(xy - xy = 0\)
- \(-9x + 7x = -2x\)
- \(z^2y\) stays as \(z^2y\)
- \(-x^2\) stays as \(-x^2\)

### Step 3: Final result
Thus, the simplified expression is:
\[
-x^2 - 2x + z^2y
\]

This is the result of adding the two polynomials.

Would you like further details or have any questions?

Here are five related questions to consider:
1. How do you subtract polynomials instead of adding them?
2. What is the importance of grouping like terms in polynomial operations?
3. Can this method be applied to polynomials with more variables?
4. How do we handle exponents when combining terms in polynomials?
5. What are some common mistakes made when adding or subtracting polynomials?

**Tip:** Always arrange your terms in decreasing order of powers (if possible) for a neater solution.

Learn how to simplify the polynomial expression (xy - 9x + z^2y) + (7x - x^2 - xy) by grouping and combining like terms. This problem covers key algebraic concepts such as polynomials and the use of the associative and commutative properties. Suitable for students in Grades 8-10.

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