We will factorize the given expression step by step:

Expression:

$a(b - c)(b + c) - d(c - b)$

Step 1: Apply difference of squares to $(b - c)(b + c)$

Using the identity $(x - y)(x + y) = x^2 - y^2$, we can rewrite $(b - c)(b + c)$ as: $(b - c)(b + c) = b^2 - c^2$

Now substitute this into the expression: $a(b^2 - c^2) - d(c - b)$

Step 2: Simplify the second term $d(c - b)$

Notice that $c - b = -(b - c)$. So we can rewrite the second term: $d(c - b) = -d(b - c)$

Thus, the expression becomes: $a(b^2 - c^2) + d(b - c)$

Step 3: Factor out common terms

We now observe that the expression has a common factor of $(b - c)$ in the second term. We can rewrite the expression as: $a(b^2 - c^2) + d(b - c) = (b - c)(a(b + c) + d)$

Final Factored Expression:

$(b - c)(a(b + c) + d)$

This is the fully factorized form of the given expression.

Would you like further explanation or examples? Here are five related questions to explore further:

1. How do you factorize polynomials with more variables?
2. What is the difference of squares identity, and when can it be used?
3. Can this expression be expanded again to verify the result?
4. How can common factors help simplify complex expressions?
5. How would this process change if negative terms were involved?

Tip: Always check if you can apply factoring identities like the difference of squares early in the factorization process—it simplifies many expressions.