Let's factor the expression \( b(ax^2 - c) + x(b^2 - ac) \).

1. **Expand the expression:**
   \[
   b(ax^2 - c) = b \cdot ax^2 - b \cdot c = abx^2 - bc
   \]
   and
   \[
   x(b^2 - ac) = x \cdot b^2 - x \cdot ac = b^2x - acx.
   \]
   So, the expression becomes:
   \[
   abx^2 - bc + b^2x - acx.
   \]

2. **Group like terms:**
   \[
   abx^2 + (b^2 - ac)x - bc.
   \]

3. **Factor by grouping:**
   - From the first two terms, \( abx^2 \) and \( (b^2 - ac)x \), factor out \( x \):
     \[
     x(abx + b^2 - ac).
     \]
   - Now the expression is:
     \[
     x(abx + b^2 - ac) - bc.
     \]
   This is the fully factored form of the expression.

Would you like more details or clarifications on any step?

Here are 5 related questions for further practice:
1. How would you factor \( ax^2 + bx + c \) in a similar context?
2. What is the distributive property, and how does it apply to factoring?
3. How do you solve a quadratic expression by factoring?
4. What is factoring by grouping, and when is it useful?
5. How can factoring help in solving polynomial equations?

**Tip:** Always check for common factors in all terms before applying more advanced factoring techniques.

Learn how to factor the expression b(ax² - c) + x(b² - ac) with a detailed step-by-step solution. This problem involves algebraic concepts like factoring by grouping and the distributive property, and is suitable for students in Grades 9-12.

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