Q:

When to use factoring by grouping for quadratic equation

There are several ways/methods to perform factoring. I am revising factoring at KhanAcademy, there are factoring by grouping, factoring special product and factoring difference of squares.
Although, I can work on the exercise, but I do not really understand when to apply factoring by grouping.

A:

Suppose your quadratic factors as $$(ax+b)(cx+d)$$ Then your quadratic was $$acx^2+(ad+bc)x+bd$$ This can be factored by grouping if you can hit upon the correct way of splitting the coefficient of $x$ into two pieces, $$acx^2+(ad+bc)x+bd=(acx^2+adx)+(bcx+bd)=ax(cx+d)+b(cx+d)=(ax+b)(cx+d)$$ In other words, any quadratic that factors can, in principle, be factored by grouping, but you have to be clever/lucky to find the right grouping. E.g., given $$210x^2+421x+210$$ there are a lot of way to split $421$ into a sum of two numbers; how do you know that the way that will work is $421=225+196$? $$210x^2+421x+210=210x^2+225x+196x+210=15x(14x+15)+14(14x+15)=(15x+14)(14x+15)$$

