I was asked to prepare a set of notes for year 12 students. These notes are going to be used by these students to prepare for the examinations. Naturally, I searched for problems to include so the students could get practice and I came across the following problem
Factorise $7x^2 + 7x - 7$ as far as possible
When making the solutions document I had difficulties in going far "as far as possible". I easily saw that $7(x^2 + x - 1)$ was one possible factorisation, but then I proceeded to see if I could factorise $x^2 + x - 1$ and it was here that I became stuck. I tried using the quadratic formula to find the roots and express $x^2 + x - 1$ as $(x_1 - root1)(x_2 - root2)$, but I couldn't seem to find such an expression. After all of that I used Wolfram Alpha and was given an irreducible factorisation of $-1/4 (-2x + \sqrt{5} -1)(2x + \sqrt{5} + 1)$. I tried to work backwards to see if could come to this factorisation, but either my lack of creativity or my terrible computation skills have left with me no solution.
If I could get a hint (maybe a solution too, but one hidden by spoiler block would be nice just so I can attempt it myself first), I'd be really thankful.