Originally Posted by
topsquark $\displaystyle 20b^2 - 11b - 3$
Multiply the coefficient of the leading term by the constant term:
20 * -3 = -60
List all the pairs of factors of -60:
1, -60
2, -30
3, -20
4, -15
5, -12
6, -10
10, -6
12, -5
15, -4
20, -3
30, -2
60, -1
Now we want to find the pair of factors in the list above that add up to be equal to the coefficient of the linear term, in this case -11. (If no such pair of factors is in the list then you cannot factor the quadratic.)
I've got 4 + -15 = -11.
So split the linear term of the quadratic into -11x = 4x - 15x.
$\displaystyle 20b^2 - 11b - 3 = 20b^2 + 4b - 15b - 3$
Group the terms as follows:
$\displaystyle = (20b^2 + 4b) + (-15b - 3)$
Now factor from each group:
$\displaystyle = 4b(5b + 1) + (-3)(5b + 1)$
and note that both terms have a common factor, 5b + 1, so we can factor that from each term as well:
$\displaystyle = (4b - 3)(5b + 1)$
There you go!
-Dan