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.