https://file.aiquickdraw.com/mathbot/file/1726139645278bvl8t79e.jpg

The image contains four polynomial expressions that involve adding or subtracting terms. Let's go through each one and simplify them.

### a. Simplify: 
\[
(8y - 7x^3 - 5x^2y) + (4x^2y + 8x^3 + 2y)
\]

Combine like terms:
- For \( y \): \( 8y + 2y = 10y \)
- For \( x^3 \): \( -7x^3 + 8x^3 = x^3 \)
- For \( x^2y \): \( -5x^2y + 4x^2y = -x^2y \)

The simplified expression:
\[
10y + x^3 - x^2y
\]

---

### b. Simplify: 
\[
(7 - 5y + 2x^3y^2) + (10x^3y^2 + 5y - 3)
\]

Combine like terms:
- For constants: \( 7 - 3 = 4 \)
- For \( y \): \( -5y + 5y = 0 \) (they cancel out)
- For \( x^3y^2 \): \( 2x^3y^2 + 10x^3y^2 = 12x^3y^2 \)

The simplified expression:
\[
4 + 12x^3y^2
\]

---

### c. Simplify: 
\[
(5m + 7m^2 - 11m^3) - (-7m^3 + 8m - 4m^2)
\]

Distribute the negative sign:
\[
5m + 7m^2 - 11m^3 + 7m^3 - 8m + 4m^2
\]

Now, combine like terms:
- For \( m \): \( 5m - 8m = -3m \)
- For \( m^2 \): \( 7m^2 + 4m^2 = 11m^2 \)
- For \( m^3 \): \( -11m^3 + 7m^3 = -4m^3 \)

The simplified expression:
\[
-3m + 11m^2 - 4m^3
\]

---

### d. Simplify: 
\[
(-10b + 7b^2 - 1) - (2b - 6 + 5b^2)
\]

Distribute the negative sign:
\[
-10b + 7b^2 - 1 - 2b + 6 - 5b^2
\]

Now, combine like terms:
- For \( b \): \( -10b - 2b = -12b \)
- For \( b^2 \): \( 7b^2 - 5b^2 = 2b^2 \)
- For constants: \( -1 + 6 = 5 \)

The simplified expression:
\[
-12b + 2b^2 + 5
\]

---

Would you like further clarification on any of these simplifications, or details on how the terms were combined?

Here are 5 related questions for further exploration:
1. What is the process for simplifying polynomial expressions involving multiple variables?
2. How do you handle negative signs when simplifying complex expressions?
3. How can factoring help in simplifying polynomial expressions?
4. What are the differences between like terms and unlike terms in polynomial operations?
5. How can these simplified expressions be applied in solving algebraic equations?

**Tip**: When simplifying expressions, always look for common terms that share the same variable and exponent—this is key to combining them correctly.

Simplify the following polynomial expressions: a) (8y - 7x^3 - 5x^2y) + (4x^2y + 8x^3 + 2y), b) (7 - 5y + 2x^3y^2) + (10x^3y^2 + 5y - 3), c) (5m + 7m^2 - 11m^3) - (-7m^3 + 8m - 4m^2), d) (-10b + 7b^2 - 1) - (2b - 6 + 5b^2)

Learn how to simplify polynomial expressions involving multiple variables. This step-by-step solution covers essential algebra concepts like combining like terms and simplifying complex polynomials, suitable for Grades 8-10.

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