Question 115652


Looking at {{{3x^2+8x+5}}} we can see that the first term is {{{3x^2}}} and the last term is {{{5}}} where the coefficients are 3 and 5 respectively.


Now multiply the first coefficient 3 and the last coefficient 5 to get 15. Now what two numbers multiply to 15 and add to the  middle coefficient 8? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 8


<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=16</td></tr><tr><td align="center">3</td><td align="center">5</td><td>3+5=8</td></tr><tr><td align="center">-1</td><td align="center">-15</td><td>-1+(-15)=-16</td></tr><tr><td align="center">-3</td><td align="center">-5</td><td>-3+(-5)=-8</td></tr></table>



From this list we can see that 3 and 5 add up to 8 and multiply to 15



Now looking at the expression {{{3x^2+8x+5}}}, replace {{{8x}}} with {{{3x+5x}}} (notice {{{3x+5x}}} adds up to {{{8x}}}. So it is equivalent to {{{8x}}})


{{{3x^2+highlight(3x+5x)+5}}}



Now let's factor {{{3x^2+3x+5x+5}}} by grouping:



{{{(3x^2+3x)+(5x+5)}}} Group like terms



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



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


So {{{3x^2+3x+5x+5}}} factors to {{{(3x+5)(x+1)}}}



So this also means that {{{3x^2+8x+5}}} factors to {{{(3x+5)(x+1)}}} (since {{{3x^2+8x+5}}} is equivalent to {{{3x^2+3x+5x+5}}})


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Answer:


So {{{3x^2+8x+5}}} factors to {{{(3x+5)(x+1)}}}