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Looking at we can see that the first term is and the last term is where the coefficients are 1 and 9 respectively.
Now multiply the first coefficient 1 and the last coefficient 9 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
First Number
Second Number
Sum
1
9
1+9=10
3
3
3+3=6
-1
-9
-1+(-9)=-10
-3
-3
-3+(-3)=-6
From this list we can see that -3 and -3 add up to -6 and multiply to 9
Now looking at the expression , replace with (notice adds up to . So it is equivalent to )
Now let's factor by grouping:
Group like terms
Factor out the GCF of out of the first group. Factor out the GCF of out of the second group
Since we have a common term of , we can combine like terms
So factors to
So this also means that factors to (since is equivalent to )
note: is equivalent to since the term occurs twice. So also factors to
(