Simplify as a fraction in lowest terms

Question

How would you simplify the above without directly calculating each part of the expression?

Taken from the 2009 IWYMIC

Answer 1

If Mark's cryptic hint isn't sufficient for you, here is another one:-

$$x^4 + x^2 + 1 = x^4 + 2x^2 + 1 - x^2 = (x^2+1)^2-x^2 = (x^2+x+1)(x^2-x+1). $$

Answer 2

Hint $$ x^3 -1 = (x-1)(x^2 + x + 1). $$ That should help you with expressions like $5^4 + 5^2 + 1$.

Answer 3

Here is a cryptic hint which will help if you see a main idea - so it still gives you a chance to puzzle something substantial out yourself.$$x^2+x+1=(x+1)^2-(x+1)+1$$