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.