Question 185047
I assume that you want to factor.



{{{36x^2+24x+4}}} Start with the given expression



{{{4(9x^2+6x+1)}}} Factor out the GCF {{{4}}}



Now let's focus on the inner expression {{{9x^2+6x+1}}}





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Looking at {{{9x^2+6x+1}}} we can see that the first term is {{{9x^2}}} and the last term is {{{1}}} where the coefficients are 9 and 1 respectively.


Now multiply the first coefficient 9 and the last coefficient 1 to get 9. Now what two numbers multiply to 9 and add to the  middle coefficient 6? Let's list all of the factors of 9:




Factors of 9:

1,3


-1,-3 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 9

1*9

3*3

(-1)*(-9)

(-3)*(-3)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">9</td><td>1+9=10</td></tr><tr><td align="center">3</td><td align="center">3</td><td>3+3=6</td></tr><tr><td align="center">-1</td><td align="center">-9</td><td>-1+(-9)=-10</td></tr><tr><td align="center">-3</td><td align="center">-3</td><td>-3+(-3)=-6</td></tr></table>



From this list we can see that 3 and 3 add up to 6 and multiply to 9



Now looking at the expression {{{9x^2+6x+1}}}, replace {{{6x}}} with {{{3x+3x}}} (notice {{{3x+3x}}} adds up to {{{6x}}}. So it is equivalent to {{{6x}}})


{{{9x^2+highlight(3x+3x)+1}}}



Now let's factor {{{9x^2+3x+3x+1}}} by grouping:



{{{(9x^2+3x)+(3x+1)}}} Group like terms



{{{3x(3x+1)+1(3x+1)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{1}}} out of the second group



{{{(3x+1)(3x+1)}}} Since we have a common term of {{{3x+1}}}, we can combine like terms


So {{{9x^2+3x+3x+1}}} factors to {{{(3x+1)(3x+1)}}}



So this also means that {{{9x^2+6x+1}}} factors to {{{(3x+1)(3x+1)}}} (since {{{9x^2+6x+1}}} is equivalent to {{{9x^2+3x+3x+1}}})



note:  {{{(3x+1)(3x+1)}}} is equivalent to  {{{(3x+1)^2}}} since the term {{{3x+1}}} occurs twice. So {{{9x^2+6x+1}}} also factors to {{{(3x+1)^2}}}




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So our expression goes from {{{4(9x^2+6x+1)}}} and factors further to {{{4(3x+1)^2}}}



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Answer:


So {{{36x^2+24x+4}}} factors to {{{4(3x+1)^2}}}