A polynomial expression will only be factorable if it crosses or touches the X-axis. Note, however, if you can use Complex (so called "imaginary") numbers then all polynomials are factorable.

What makes a polynomial not Factorable?

The most reliable way I can think of to find out if a polynomial is factorable or not is to plug it into your calculator, and find your zeroes. If those zeroes are weird long decimals (or don't exist), then you probably can't factor it. Then, you'd have to use the quadratic formula.

Can some polynomials not be factored?

A polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial .