You can simplify polynomials only if they have roots. You can think of polynomials as numbers, and of monomials of the form (x-a) as prime numbers. So, as you can write a composite numbers as product of primes, you can write a "composite" polynomial as product of monomials of the form (x-a), where a is a root of the polynomial. If the polynomial has no roots, it means that, in a certain sense, it is "prime", and cannot thus be further simplified.
For example, x^2+1 has no (real) roots, so it cannot be simplified. On the other hand, x^2-1 has roots \pm1, so it can be simplified into x(+1)(x-1).
Finally, x^3+x has a root for x=0. So, we can write as x(x^2+1), and for what we saw before, this expression is no longer simplifiable.
