Question 167732


{{{3ax^2+7ax+2a}}} Start with the given expression



{{{a(3x^2+7x+2)}}} Factor out the GCF {{{a}}}



Now let's focus on the inner expression {{{3x^2+7x+2}}}





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Looking at {{{3x^2+7x+2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{2}}} where the coefficients are 3 and 2 respectively.


Now multiply the first coefficient 3 and the last coefficient 2 to get 6. Now what two numbers multiply to 6 and add to the  middle coefficient 7? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7


<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=7</td></tr><tr><td align="center">2</td><td align="center">3</td><td>2+3=5</td></tr><tr><td align="center">-1</td><td align="center">-6</td><td>-1+(-6)=-7</td></tr><tr><td align="center">-2</td><td align="center">-3</td><td>-2+(-3)=-5</td></tr></table>



From this list we can see that 1 and 6 add up to 7 and multiply to 6



Now looking at the expression {{{3x^2+7x+2}}}, replace {{{7x}}} with {{{1x+6x}}} (notice {{{1x+6x}}} adds up to {{{7x}}}. So it is equivalent to {{{7x}}})


{{{3x^2+highlight(1x+6x)+2}}}



Now let's factor {{{3x^2+1x+6x+2}}} by grouping:



{{{(3x^2+1x)+(6x+2)}}} Group like terms



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



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


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



So this also means that {{{3x^2+7x+2}}} factors to {{{(x+2)(3x+1)}}} (since {{{3x^2+7x+2}}} is equivalent to {{{3x^2+1x+6x+2}}})




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So our expression goes from {{{a(3x^2+7x+2)}}} and factors further to {{{a(x+2)(3x+1)}}}



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Answer:


So {{{3ax^2+7ax+2a}}} factors to {{{a(x+2)(3x+1)}}}