Question 116031

{{{-10w^2-28w+6}}} Start with the given expression



{{{-2(5w^2+14w-3)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{5w^2+14w-3}}}





Looking at {{{5w^2+14w-3}}} we can see that the first term is {{{5w^2}}} and the last term is {{{-3}}} where the coefficients are 5 and -3 respectively.


Now multiply the first coefficient 5 and the last coefficient -3 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient 14? Let's list all of the factors of -15:




Factors of -15:

1,3,5,15


-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -15

(1)*(-15)

(3)*(-5)

(-1)*(15)

(-3)*(5)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 14


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-15</td><td>1+(-15)=-14</td></tr><tr><td align="center">3</td><td align="center">-5</td><td>3+(-5)=-2</td></tr><tr><td align="center">-1</td><td align="center">15</td><td>-1+15=14</td></tr><tr><td align="center">-3</td><td align="center">5</td><td>-3+5=2</td></tr></table>



From this list we can see that -1 and 15 add up to 14 and multiply to -15



Now looking at the expression {{{5w^2+14w-3}}}, replace {{{14w}}} with {{{-1w+15w}}} (notice {{{-1w+15w}}} adds up to {{{14w}}}. So it is equivalent to {{{14w}}})


{{{5w^2+highlight(-1w+15w)+-3}}}



Now let's factor {{{5w^2-1w+15w-3}}} by grouping:



{{{(5w^2-1w)+(15w-3)}}} Group like terms



{{{w(5w-1)+3(5w-1)}}} Factor out the GCF of {{{w}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group



{{{(w+3)(5w-1)}}} Since we have a common term of {{{5w-1}}}, we can combine like terms


So {{{5w^2-1w+15w-3}}} factors to {{{(w+3)(5w-1)}}}



So this also means that {{{5w^2+14w-3}}} factors to {{{(w+3)(5w-1)}}} (since {{{5w^2+14w-3}}} is equivalent to {{{5w^2-1w+15w-3}}})



{{{-2(w+3)(5w-1)}}} Now reintroduce the GCF


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Answer:


So {{{-10w^2-28w+6}}} factors to {{{-2(w+3)(5w-1)}}}