Question 339213
I'm assuming you want to factor.



Looking at the expression {{{x^2+10xy+16y^2}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{10}}}, and the last coefficient is {{{16}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{16}}} to get {{{(1)(16)=16}}}.



Now the question is: what two whole numbers multiply to {{{16}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{10}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{16}}} (the previous product).



Factors of {{{16}}}:

1,2,4,8,16

-1,-2,-4,-8,-16



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{16}}}.

1*16 = 16
2*8 = 16
4*4 = 16
(-1)*(-16) = 16
(-2)*(-8) = 16
(-4)*(-4) = 16


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{10}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>1+16=17</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>8</font></td><td  align="center"><font color=red>2+8=10</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>4+4=8</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-16</font></td><td  align="center"><font color=black>-1+(-16)=-17</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-2+(-8)=-10</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-4+(-4)=-8</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{8}}} add to {{{10}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{8}}} both multiply to {{{16}}} <font size=4><b>and</b></font> add to {{{10}}}



Now replace the middle term {{{10xy}}} with {{{2xy+8xy}}}. Remember, {{{2}}} and {{{8}}} add to {{{10}}}. So this shows us that {{{2xy+8xy=10xy}}}.



{{{x^2+highlight(2xy+8xy)+16y^2}}} Replace the second term {{{10xy}}} with {{{2xy+8xy}}}.



{{{(x^2+2xy)+(8xy+16y^2)}}} Group the terms into two pairs.



{{{x(x+2y)+(8xy+16y^2)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+2y)+8y(x+2y)}}} Factor out {{{8y}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(x+8y)(x+2y)}}} Combine like terms. Or factor out the common term {{{x+2y}}}



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Answer:



So {{{x^2+10xy+16y^2}}} factors to {{{(x+8y)(x+2y)}}}.



In other words, {{{x^2+10xy+16y^2=(x+8y)(x+2y)}}}.



Note: you can check the answer by expanding {{{(x+8y)(x+2y)}}} to get {{{x^2+10xy+16y^2}}} or by graphing the original expression and the answer (the two graphs should be identical).



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Jim