How do you know if a quadratic equation can be factored?

The answer depends on what the factors will be. For example, every quadratic can be factored if you allow complex numbers.

If not, then it helps to use the discriminant.

If it is positive, there are two real factors or solutions.

If that positive number is a perfect square, then the factors are rational numbers.

If not, they are real but not rational (irrational).

If the discriminant is 0, there is one real solution.

Lastly, if it is negative, there are no real solutions.