Using Complex Numbers In Factorization

Being able to factorize a polynomial into a product of smaller polynomials is incredibly powerful tool to find their solutions. Simply put, we can factorize,

In this case, the expression was real and its two factors were also real.

But most expressions are not neatly factorizable with simple algebraic formulas. Expressions such as $a^2 + b^2$ and more do not have real factors. In some of these cases, we can use complex numbers to factorize real expressions into some complex factors. Usually this is done through replacing $i^2$ with $-1$. Explore the questions below on how such factorization can happen.