Subtract (4x square-5xy+7y) from the sum of (2x square-9x y square+5y)and (6x square-4 x square y+4y)

To solve this problem, we need to follow the steps sequentially.

### Step 1: Find the sum of the two expressions
The expressions to add are:

\[
(2x^2 - 9xy^2 + 5y) + (6x^2 - 4x^2y + 4y)
\]

Combine the like terms:

- Combine the \(x^2\) terms: \(2x^2 + 6x^2 = 8x^2\)
- Combine the \(xy^2\) terms: \(-9xy^2\) (there's no like term to add here)
- Combine the \(x^2y\) terms: \(-4x^2y\) (there's no like term to add here)
- Combine the \(y\) terms: \(5y + 4y = 9y\)

So, the sum is:

\[
8x^2 - 9xy^2 - 4x^2y + 9y
\]

### Step 2: Subtract the expression \((4x^2 - 5xy + 7y)\) from the result
Now, subtract \((4x^2 - 5xy + 7y)\) from the sum we just found:

\[
(8x^2 - 9xy^2 - 4x^2y + 9y) - (4x^2 - 5xy + 7y)
\]

Distribute the negative sign to each term in the second expression:

\[
8x^2 - 9xy^2 - 4x^2y + 9y - 4x^2 + 5xy - 7y
\]

### Step 3: Combine like terms
Combine the like terms in the expression:

- Combine the \(x^2\) terms: \(8x^2 - 4x^2 = 4x^2\)
- Combine the \(xy^2\) term: \(-9xy^2\) (no like term)
- Combine the \(x^2y\) term: \(-4x^2y\) (no like term)
- Combine the \(xy\) terms: \(5xy\) (no like term)
- Combine the \(y\) terms: \(9y - 7y = 2y\)

So, the final expression is:

\[
\boxed{4x^2 - 9xy^2 - 4x^2y + 5xy + 2y}
\]

This is the simplified expression after performing the required operations.

Would you like any more details or have any questions?

### Related Questions:
1. How do you identify and combine like terms in a polynomial?
2. What are the properties of subtraction in algebra?
3. Can you factor the final expression further?
4. How does changing the sign of terms affect the subtraction process?
5. What is the difference between subtraction and addition of polynomials?

### Tip:
Always be careful with the signs when subtracting expressions, especially when distributing the negative sign across multiple terms.

Learn how to subtract polynomials step-by-step. Follow the detailed solution to subtract (4x^2 - 5xy + 7y) from the sum of (2x^2 - 9xy^2 + 5y) and (6x^2 - 4x^2y + 4y), including combining like terms and distributing negative signs. Suitable for Grades 7-9.

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