Question 513466
First, let's substitute in for b, c, and d, since we are given formulae for all three in terms of a:

{{{ x^a + x^b + x^c + x^d = x^a + x^(a + 2) + x^(a + 4) + x^(a + 5) }}}

Now let's remember how exponents add:

{{{ x^(p + q) = x^p * x^q }}}

for all p and q.  Thus we can rewrite our expression as:

{{{ x^a + x^a * x^2 + x^a * x^4 + x^a * x^5 }}}

Now let's look for ways to factor our expression.  Notice that x^a is a common factor in each term, so we'll factor x^a out:

{{{ x^a * (1 + x^2 + x^4 + x^5) }}}

There doesn't appear to be any way to easily factor the remaining polynomial (the only possibilities for rational roots are 1 and -1, and neither of these work), so it looks like we're done.