Factor the given equation: b(ax² - c) + X(b²-ac) 

To factor the given expression \( b(ax^2 - c) + X(b^2 - ac) \), let's follow these steps:

1. **Distribute** the terms within the expression:
   \[
   = b \cdot ax^2 - b \cdot c + X \cdot b^2 - X \cdot ac
   \]
   Which simplifies to:
   \[
   = abx^2 - bc + Xb^2 - Xac
   \]

2. **Group the terms** to identify common factors:
   \[
   = abx^2 - Xac + Xb^2 - bc
   \]
   Notice that the expression can be regrouped as:
   \[
   = abx^2 - Xac + Xb^2 - bc = (abx^2 - Xac) + (Xb^2 - bc)
   \]

3. **Factor out** the common factors from each group:
   - From the first group \(abx^2 - Xac\), factor out \(a\):
     \[
     = a(bx^2 - Xc)
     \]
   - From the second group \(Xb^2 - bc\), factor out \(b\):
     \[
     = b(Xb - c)
     \]

4. **Combine the factored terms**:
   \[
   = a(bx^2 - Xc) + b(Xb - c)
   \]
   Notice that this expression is already simplified. To check if it can be factored further, observe the terms:
   \[
   = a(bx^2 - Xc) + b(Xb - c)
   \]
   Since there are no common factors across the terms, the expression is factored as:
   \[
   abx^2 - Xac + Xb^2 - bc
   \]

Thus, the factorization of \( b(ax^2 - c) + X(b^2 - ac) \) yields:

\[
abx^2 - Xac + Xb^2 - bc = a(bx^2 - Xc) + b(Xb - c)
\]

Would you like more details on any specific step, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How would you factor a quadratic expression, such as \(ax^2 + bx + c\)?
2. What are the key steps in factoring by grouping?
3. Can you apply the factorization process to cubic polynomials?
4. How do you factor a difference of squares, such as \(a^2 - b^2\)?
5. What methods can be used to solve a polynomial equation once it is factored?

**Tip:** Always look for common factors in each term to simplify expressions before attempting to factor the entire equation.

Factor the given equation: b(ax² - c) + X(b² - ac)

Explore how to factor the algebraic expression b(ax² - c) + X(b² - ac) using factoring by grouping and identifying common factors. This step-by-step guide covers key algebraic concepts, including the distributive property and factoring techniques. Suitable for high school students in Grades 9-12.

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