https://file.aiquickdraw.com/mathbot/file/17288876508908jewvp4u.jpg

The expression in the image is:

\[
5(3x - 4)^2(1 - 5x)^{-3} + 3(3x - 4)(1 - 5x)^{-2}
\]

The goal is to factor this expression.

### Step-by-Step Solution:

1. **Identify common factors**: 
   Both terms in the expression contain factors involving \((3x - 4)\) and \((1 - 5x)\), but with different exponents.

   The common factors are \((3x - 4)\) and \((1 - 5x)\), and we will factor out the **smallest powers** of these terms:
   - The lowest power of \((3x - 4)\) is \( (3x - 4)^1 \).
   - The lowest power of \((1 - 5x)\) is \( (1 - 5x)^{-3} \).

2. **Factor out the common terms**: 
   We factor out \( (3x - 4)(1 - 5x)^{-3} \).

\[
(3x - 4)(1 - 5x)^{-3} \left[ 5(3x - 4) + 3(1 - 5x) \right]
\]

3. **Simplify the remaining terms inside the brackets**:
   - First, expand \( 5(3x - 4) \):
     \[
     5(3x - 4) = 15x - 20
     \]
   - Now, expand \( 3(1 - 5x) \):
     \[
     3(1 - 5x) = 3 - 15x
     \]
   - Combine these results:
     \[
     (15x - 20) + (3 - 15x) = -17
     \]

4. **Write the final factored expression**:
   The factored form of the expression is:

   \[
   (3x - 4)(1 - 5x)^{-3}(-17)
   \]

### Final Answer:
\[
-17(3x - 4)(1 - 5x)^{-3}
\]

Would you like more details on any part of this factorization or have any other questions?

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Here are 5 related questions to expand on this topic:
1. How would the factorization change if the powers of \((3x - 4)\) and \((1 - 5x)\) were different?
2. Can this expression be simplified further for specific values of \(x\)?
3. How do negative exponents affect the factoring process?
4. What would happen if the two terms had different coefficients or powers?
5. Could we rewrite the expression without using negative exponents?

### Tip:
Always look for the greatest common factor (GCF) when factoring complex expressions—it simplifies the process significantly.

Factor the expression: 5(3x - 4)^2(1 - 5x)^{-3} + 3(3x - 4)(1 - 5x)^{-2}

Learn how to factor the algebraic expression 5(3x - 4)^2(1 - 5x)^{-3} + 3(3x - 4)(1 - 5x)^{-2} using common factors and exponent rules. This step-by-step guide explains how to simplify and solve this problem for students in Grades 10-12.

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