<PRE><B>Question 114381</B>
{{{9x^2+12xy+4y^2-25}}} Start with the given expression



First let&#39;s factor the trinomial {{{9x^2+12xy+4y^2}}}:



Looking at {{{9x^2+12xy+4y^2}}} we can see that the first term is {{{9x^2}}} and the last term is {{{4y^2}}} where the coefficients are 9 and 4 respectively.


Now multiply the first coefficient 9 and the last coefficient 4 to get 36. Now what two numbers multiply to 36 and add to the  middle coefficient 12? Let&#39;s list all of the factors of 36:




Factors of 36:

1,2,3,4,6,9,12,18


-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 36

1*36

2*18

3*12

4*9

6*6

(-1)*(-36)

(-2)*(-18)

(-3)*(-12)

(-4)*(-9)

(-6)*(-6)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 12


&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;36&lt;/td&gt;&lt;td&gt;1+36=37&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;18&lt;/td&gt;&lt;td&gt;2+18=20&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;12&lt;/td&gt;&lt;td&gt;3+12=15&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;4&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;9&lt;/td&gt;&lt;td&gt;4+9=13&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;6&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;6&lt;/td&gt;&lt;td&gt;6+6=12&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-36&lt;/td&gt;&lt;td&gt;-1+(-36)=-37&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-18&lt;/td&gt;&lt;td&gt;-2+(-18)=-20&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-12&lt;/td&gt;&lt;td&gt;-3+(-12)=-15&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-4&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-9&lt;/td&gt;&lt;td&gt;-4+(-9)=-13&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-6&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-6&lt;/td&gt;&lt;td&gt;-6+(-6)=-12&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that 6 and 6 add up to 12 and multiply to 36



Now looking at the expression {{{9x^2+12xy+4y^2}}}, replace {{{12xy}}} with {{{6xy+6xy}}} (notice {{{6xy+6xy}}} adds up to {{{12xy}}}. So it is equivalent to {{{12xy}}})


{{{9x^2+highlight(6xy+6xy)+4y^2}}}



Now let&#39;s factor {{{9x^2+6xy+6xy+4y^2}}} by grouping:



{{{(9x^2+6xy)+(6xy+4y^2)}}} Group like terms



{{{3x(3x+2y)+2y(3x+2y)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{2y}}} out of the second group



{{{(3x+2y)(3x+2y)}}} Since we have a common term of {{{3x+2y}}}, we can combine like terms


So {{{9x^2+12xy+4y^2}}} factors to {{{(3x+2y)(3x+2y)}}} which is also {{{(3x+2y)^2}}}



So we then get



{{{(3x+2y)^2-25}}}



{{{(3x+2y)^2-5^2}}} Now write 25 as {{{5^2}}}




{{{(3x+2y+5)(3x+2y-5)}}} Now factor using the difference of squares


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