A normal person would factor $2x^4 - 15 x^2 -27$ by treating $x^2$ as the variable. I'm grading a test now and the student has come up with the extremely clever idea to add and subtract $6x^3 + 3x^2 + 9x$ which makes the factoring run like this:
$$2x^4+6x^3+3x^2+9x-6x^3-18x^2 - 9x -27$$
$$= x(2x^3+6x^2+3x + 9) - 3(2x^3+6x^2+3x + 9)$$
$$= (x-3)(2x^3+6x^2+3x + 9).$$
The second polynomial also factors further. How on earth could an 8th-grade B+ student come up with this scheme," I wondered. So I typed "factor $2x^4 - 15 x^2 -27$" into Google and the very first hit shows using exactly this method. I'm 100% sure this young lady (virtual student supposedly supervised by her mother) googled every answer on her test, including this one. But I was stunned that THIS was the Google answer.
So finally my question: Is there some known technique for cooking up the thing she added and subtracted? (And why would Google pass over the obvious method...?)