The answer can be yes or no depending on what you mean by factoring. If you mean using normal (real) numbers the answer is no; if you want to include complex numbers (which have an imaginary component) the answer is yes. I assume you mean ordinary numbers. Every quadratic can be graphed. It’s a parabola. If the parabola intersects the x axis, then it can be factorised. If it sits above or below the x axis it can’t be factorised. This is because the x intercepts are the zeroes of the quadratic and the zeroes are directly related to the factors.
Another meaning to factorisation is whether the factors are irrational or rational, that is, do they contain a square root? If they do, you may want to reject them as factors because irrationals cannot be precisely defined as “normal” numbers, so the answer is no, the quadratic can’t be solved. If they don’t contain square roots, the answer is yes, the quadratic can be solved.
Is every quadratic factorisable? Simple answer, no, if you stick with simple numbers, avoid irrationals, avoid complex numbers.