https://file.aiquickdraw.com/mathbot/file/17280038854331pxxjf6w.jpg

The expression you uploaded is:

\[
a^2b^2 - 2ab - 15
\]

This appears to be a quadratic-like expression with respect to the variable \(ab\). To factor it, we can treat \(ab\) as a single variable.

Let’s go through the factorization process step by step:

1. Consider substituting \( x = ab \), so the expression becomes:
   \[
   x^2 - 2x - 15
   \]
   
2. Now, factor this quadratic expression:
   \[
   x^2 - 2x - 15 = (x - 5)(x + 3)
   \]
   
3. Substitute back \( x = ab \) into the factors:
   \[
   (ab - 5)(ab + 3)
   \]

Thus, the factored form of the original expression is:

\[
(a^2b^2 - 2ab - 15) = (ab - 5)(ab + 3)
\]

Would you like further explanation, or do you have any questions?

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Here are five related questions that expand on this process:
1. How do you factor quadratics in general?
2. Can the method of substitution be applied to other forms of polynomials?
3. What is the role of the discriminant in factoring quadratics?
4. How can we verify if the factorization is correct?
5. How does factoring help in solving equations?

**Tip:** Always try substituting complex parts of an expression with a simpler variable to make factorization easier.

Learn how to factor the quadratic-like expression a^2b^2 - 2ab - 15 step by step, using substitution and standard factoring techniques. This problem introduces key algebraic methods, making it suitable for students in Grades 9-12.

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