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.
