<PRE><B>Question 112031</B>


Looking at {{{6h^2+17hk+10k^2}}} we can see that the first term is {{{6h^2}}} and the last term is {{{10k^2}}} where the coefficients are 6 and 10 respectively.


Now multiply the first coefficient 6 and the last coefficient 10 to get 60. Now what two numbers multiply to 60 and add to the  middle coefficient 17? Let&#39;s list all of the factors of 60:




Factors of 60:

1,2,3,4,5,6,10,12,15,20,30,60


-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 60

1*60

2*30

3*20

4*15

5*12

6*10

(-1)*(-60)

(-2)*(-30)

(-3)*(-20)

(-4)*(-15)

(-5)*(-12)

(-6)*(-10)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17


&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;60&lt;/td&gt;&lt;td&gt;1+60=61&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;30&lt;/td&gt;&lt;td&gt;2+30=32&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;20&lt;/td&gt;&lt;td&gt;3+20=23&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;15&lt;/td&gt;&lt;td&gt;4+15=19&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;5&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;12&lt;/td&gt;&lt;td&gt;5+12=17&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;10&lt;/td&gt;&lt;td&gt;6+10=16&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;-60&lt;/td&gt;&lt;td&gt;-1+(-60)=-61&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;-30&lt;/td&gt;&lt;td&gt;-2+(-30)=-32&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;-20&lt;/td&gt;&lt;td&gt;-3+(-20)=-23&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;-15&lt;/td&gt;&lt;td&gt;-4+(-15)=-19&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-5&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-12&lt;/td&gt;&lt;td&gt;-5+(-12)=-17&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;-10&lt;/td&gt;&lt;td&gt;-6+(-10)=-16&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that 5 and 12 add up to 17 and multiply to 60



Now looking at the expression {{{6h^2+17hk+10k^2}}}, replace {{{17hk}}} with {{{5hk+12hk}}} (notice {{{5hk+12hk}}} adds up to {{{17hk}}}. So it is equivalent to {{{17hk}}})


{{{6h^2+highlight(5hk+12hk)+10k^2}}}



Now let&#39;s factor {{{6h^2+5hk+12hk+10k^2}}} by grouping:



{{{(6h^2+5hk)+(12hk+10k^2)}}} Group like terms



{{{h(6h+5k)+2k(6h+5k)}}} Factor out the GCF of {{{h}}} out of the first group. Factor out the GCF of {{{2k}}} out of the second group



{{{(h+2k)(6h+5k)}}} Since we have a common term of {{{6h+5k}}}, we can combine like terms


So {{{6h^2+5hk+12hk+10k^2}}} factors to {{{(h+2k)(6h+5k)}}}



So this also means that {{{6h^2+17hk+10k^2}}} factors to {{{(h+2k)(6h+5k)}}} (since {{{6h^2+17hk+10k^2}}} is equivalent to {{{6h^2+5hk+12hk+10k^2}}})
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