Question 22569
Factoring trinomials. 
N.B. You can indicate a square by using the caret (^), thus...x^2 = x-squared.
Example, factor:

{{{b^2 + 20b + 36 = 0}}} 

When you factor a trinomial such as the one above, you are looking for two binomials that will look like this: (b + m)(b + n) and when you multiply these binomial factors, you will get back the trinomial you started with.

{{{b^2 + 20b + 36 = (b + m)(b + n)}}} So, what are m and n and how do you find them?

Let's multiply the two binomials (b + m)(b + n) and compare the result with the original trinomial.

{{{(b + m)(b + n) = b^2 + nb + mb + mn)}}} Simplify. 
{{{b^2 + (m+n)b + mn}}} compare this with:
{{{b^2 + 20b + 36}}}

You can see that for these two trinomials to be equal, then (m+n) = 20 and mn = 36. That is, you are looking for two integers whose sum is 20 and whose product is 36. Let's try some examples:

4 X 9 = 36 and 4 + 9 = 13  No good!
6 X 6 = 36 and 6 + 6 = 12  No good!
2 X 18 = 36 and 2 + 18 = 20 Bingo! So, m = 2 and n = 18

So, mn = 2(18) = 36 and m+n = 2+18 = 20

Now you can write your two binomial factors (b + m)(b + n) as:
{{{(b + 2)(b + 18) = b^2 + 20b + 36}}}