Question 250595
{{{16x^2 + 24X + 9}}}

To factor this, you sort of do a "reverse" of FOIL.  You know that FOIL stands for First, Outside, Inside, Last.  

Usually with FOIL, we start with two sets of parenthesis:

We want the first term in each set of parenthesis to multiply together to reach {{{16x^2}}}.


So, let's think of a fairly typical way that you can reach 16 when you multiply two numbers together.  How about the number 4?  


4 times 4 = 16 yes?  We cannot forget there is an {{{x^2}}} term there, so let's put in that x times x = {{{x^2}}}.


We now have:

(4X ________)(4X ________)


Now in the equation you gave: {{{16x^2 + 24X + 9}}}  we have to come up with a way that will let us MULTIPLY two numbers to reach 9 and yet ADD together to reach 24.   


A pretty typical way to reach 9 thru multiplication of two numbers is to use 3 times 3.


So, now we have:


(4X ?????  3)(4x  ?????   3).


What do we do with the 3?  Is it positive or negative?  Well the 9 in the equation is positive, as is the 24, so let's try a positive 3.......


(4x + 3)(4x + 3)


Does this give us our original equation of:  {{{16x^2 + 24X + 9}}}?


Let's check:

FOIL:

First terms:  4x times 4x = {{{16x^2}}}
Outside terms:  4x times 3 = 12x
Inside terms:   4x times 3 = 12x
Last terms:  3 times 3 = 9



That means we have:  {{{16x^2}}} + 12x + 12x + 9

Let's combine like terms of 12x + 12x = 24X and rewrite what we have:  {{{16x^2}}} + 24x + 9  


Ok! That's our original equation.  Therefore, (4x + 3)(4x + 3) 


(or you can write it as:  {{{(4x + 3)^2}}}) is a way to factor:  {{{16x^2}}} + 24x = 9  


I hope this helps. :-)