Answer
First, find the product of $a$ and $c$, as long as $a\ne1$.
Then, we look for two numbers whose sum is $b$ and whose product is $a*c$.
Then, we rewrite the polynomial with two blanks.
Since we would want (usually) a positive pair of numbers, we would then look at factors of $a*c$ and look at their sums.
Then, replace the blanks with the pair of factors that have the correct middle term. Now you can factor by grouping (and factoring out common terms).
Work Step by Step
Example: $2x^2+6x+4$
$a*c = 2*4=8$
$b=6$
(We want a pair of numbers whose product is 8 and sum is 6.)
$2x^2+ax+bx+4$
Factors of 8: 1 and 8, 2 and 4
Sums of factors of 8:
$1+8=9$
$2+4=6$
$2x^2+2x+4x+4$
$2x(x+1)+4(x+1)$
$(2x+4)(x+1)$
$2(x+2)(x+1)$
