To solve the expression 

\[
\frac{12a^3 + 33b^2 - 35a^2b - 10b^3}{4a - 5b}
\]

we will perform polynomial long division to divide the numerator by the denominator.

### Step-by-Step Solution

1. **Divide the leading term of the numerator by the leading term of the denominator:**

   The leading term of the numerator is \(12a^3\), and the leading term of the denominator is \(4a\).

   \[
   \frac{12a^3}{4a} = 3a^2
   \]

2. **Multiply the entire denominator by this result:**

   \[
   (4a - 5b) \cdot 3a^2 = 12a^3 - 15a^2b
   \]

3. **Subtract this from the original numerator:**

   \[
   (12a^3 + 33b^2 - 35a^2b - 10b^3) - (12a^3 - 15a^2b) = (-35a^2b + 33b^2) - (-15a^2b)
   \]

   Simplify:

   \[
   -20a^2b + 33b^2
   \]

4. **Repeat the process with the new polynomial:**

   Divide the new leading term by the leading term of the denominator:

   \[
   \frac{-20a^2b}{4a} = -5ab
   \]

5. **Multiply the entire denominator by this result:**

   \[
   (4a - 5b) \cdot (-5ab) = -20a^2b + 25b^2
   \]

6. **Subtract this from the current polynomial:**

   \[
   (-20a^2b + 33b^2) - (-20a^2b + 25b^2) = 8b^2
   \]

7. **Divide again by the leading term:**

   \[
   \frac{8b^2}{4a} = 2b
   \]

8. **Multiply the entire denominator by this result:**

   \[
   (4a - 5b) \cdot 2b = 8b^2 - 10b^3
   \]

9. **Subtract this from the new polynomial:**

   \[
   (8b^2 - 10b^3) - (8b^2 - 10b^3) = 0
   \]

So, the division results in:

\[
3a^2 - 5ab + 2b
\]

### Final Answer

\[
\frac{12a^3 + 33b^2 - 35a^2b - 10b^3}{4a - 5b} = 3a^2 - 5ab + 2b
\]

Would you like to go deeper into any part of the solution or ask any related questions? Here are some related questions you might find interesting:

1. What is polynomial long division, and why does it work?
2. How do you factor polynomials to simplify them before division?
3. What are the Remainder and Factor Theorems in algebra?
4. How do you perform synthetic division for polynomials?
5. Can this polynomial division be used to solve real-world problems?

**Tip:** Always double-check each step in a polynomial division to avoid calculation mistakes!

12a^3 + 33b^2 - 35a^2b - 10b^3 divided by 4a - 5b

Learn how to divide the polynomial 12a^3 + 33b^2 - 35a^2b - 10b^3 by 4a - 5b using polynomial long division. This step-by-step solution covers algebraic concepts including polynomial division, subtraction of polynomials, and handling complex terms. Suitable for students in Grades 10-12.

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