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x² + 9x + 1
let's start by grouping the ones with the same variable
(x² + 9x) + 1
(x² + 9x + [?]²) + 1
so, we seem to be missing a value there, to get a perfect square trinomial, let's recall a perfect square trinomial has a middle term that is 2 * "other two values", namely
so, the middle term on this group will be 9x.
we know 2*x*[?] = 9x, so then
so that's our mystery felllow.
now, let's bear in mind that we'll be borrowing from our very good friend Mr Zero, 0, so if we add (9/2)², we also have to subtract (9/2)².