Question 139593

Looking at {{{5x^2+16x+3}}} we can see that the first term is {{{5x^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 16? 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 16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 16


<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 1 and 15 add up to 16 and multiply to 15



Now looking at the expression {{{5x^2+16x+3}}}, replace {{{16x}}} with {{{1x+15x}}} (notice {{{1x+15x}}} adds up to {{{16x}}}. So it is equivalent to {{{16x}}})


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



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



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



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



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


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



So this also means that {{{5x^2+16x+3}}} factors to {{{(x+3)(5x+1)}}} (since {{{5x^2+16x+3}}} is equivalent to {{{5x^2+1x+15x+3}}})



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Answer:


So {{{5x^2+16x+3}}} factors to {{{(x+3)(5x+1)}}}