Question 405580

{{{10a^2+25a-15}}} Start with the given expression



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



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





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


Now multiply the first coefficient 2 and the last coefficient -3 to get -6. Now what two numbers multiply to -6 and add to the  middle coefficient 5? Let's list all of the factors of -6:




Factors of -6:

1,2,3,6


-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -6

(1)*(-6)

(2)*(-3)

(-1)*(6)

(-2)*(3)


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



Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-6</td><td>1+(-6)=-5</td></tr><tr><td align="center">2</td><td align="center">-3</td><td>2+(-3)=-1</td></tr><tr><td align="center">-1</td><td align="center">6</td><td>-1+6=5</td></tr><tr><td align="center">-2</td><td align="center">3</td><td>-2+3=1</td></tr></table>



From this list we can see that -1 and 6 add up to 5 and multiply to -6



Now looking at the expression {{{2a^2+5a-3}}}, replace {{{5a}}} with {{{-a+6a}}} (notice {{{-a+6a}}} adds up to {{{5a}}}. So it is equivalent to {{{5a}}})


{{{2a^2+highlight(-a+6a)-3}}}



Now let's factor {{{2a^2-a+6a-3}}} by grouping:



{{{(2a^2-a)+(6a-3)}}} Group like terms



{{{a(2a-1)+3(2a-1)}}} Factor out the GCF of {{{a}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group



{{{(a+3)(2a-1)}}} Since we have a common term of {{{2a-1}}}, we can combine like terms


So {{{2a^2-a+6a-3}}} factors to {{{(a+3)(2a-1)}}}



So this also means that {{{2a^2+5a-3}}} factors to {{{(a+3)(2a-1)}}} (since {{{2a^2+5a-3}}} is equivalent to {{{2a^2-a+6a-3}}})




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So our expression goes from {{{5(2a^2+5a-3)}}} and factors further to {{{5(a+3)(2a-1)}}}



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Answer:


So {{{10a^2+25a-15}}} factors to {{{5(a+3)(2a-1)}}}

    

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