Question 422731
I'm assuming you want to factor this.





{{{2y^2+20y+18}}} Start with the given expression.



{{{2(y^2+10y+9)}}} Factor out the GCF {{{2}}}.



Now let's try to factor the inner expression {{{y^2+10y+9}}}



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Looking at the expression {{{y^2+10y+9}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{10}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{9}}} to get {{{(1)(9)=9}}}.



Now the question is: what two whole numbers multiply to {{{9}}} (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 {{{9}}} (the previous product).



Factors of {{{9}}}:

1,3,9

-1,-3,-9



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



These factors pair up and multiply to {{{9}}}.

1*9 = 9
3*3 = 9
(-1)*(-9) = 9
(-3)*(-3) = 9


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=red>1</font></td><td  align="center"><font color=red>9</font></td><td  align="center"><font color=red>1+9=10</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>3+3=6</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>-1+(-9)=-10</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-3+(-3)=-6</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{9}}} add to {{{10}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{9}}} both multiply to {{{9}}} <font size=4><b>and</b></font> add to {{{10}}}



Now replace the middle term {{{10y}}} with {{{y+9y}}}. Remember, {{{1}}} and {{{9}}} add to {{{10}}}. So this shows us that {{{y+9y=10y}}}.



{{{y^2+highlight(y+9y)+9}}} Replace the second term {{{10y}}} with {{{y+9y}}}.



{{{(y^2+y)+(9y+9)}}} Group the terms into two pairs.



{{{y(y+1)+(9y+9)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y+1)+9(y+1)}}} Factor out {{{9}}} 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.



{{{(y+9)(y+1)}}} Combine like terms. Or factor out the common term {{{y+1}}}



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So {{{2(y^2+10y+9)}}} then factors further to {{{2(y+9)(y+1)}}}



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Answer:



So {{{2y^2+20y+18}}} completely factors to {{{2(y+9)(y+1)}}}.



In other words, {{{2y^2+20y+18=2(y+9)(y+1)}}}.



Note: you can check the answer by expanding {{{2(y+9)(y+1)}}} to get {{{2y^2+20y+18}}} or by graphing the original expression and the answer (the two graphs should be identical).



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