Are all quadratics factorable into a product of two binomials? I'm learning algebra in school, and my teacher said that all quadratics are factorable into a product of two binomials. I then realized however that some quadratics would have imaginary roots, and therefore wouldn't be able to be put into factored form. Who's wrong here, my teacher or me? For example, can $x^2 + 4x + 1$ even be expressed in factored form? Thanks in advance.

It really depends on whether you want to have complex factors or not. If you can have complex factors, every expression can be.  If not, then only if $b^2\ge4ac$ would they be factorable.
Take $x^2+1$, it can be factored into $(x-i)(x+i)$ but none of the factors are real.
