Question 115002


Looking at {{{2p^2+11p+12}}} we can see that the first term is {{{2p^2}}} and the last term is {{{12}}} where the coefficients are 2 and 12 respectively.


Now multiply the first coefficient 2 and the last coefficient 12 to get 24. Now what two numbers multiply to 24 and add to the  middle coefficient 11? Let's list all of the factors of 24:




Factors of 24:

1,2,3,4,6,8,12,24


-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 24

1*24

2*12

3*8

4*6

(-1)*(-24)

(-2)*(-12)

(-3)*(-8)

(-4)*(-6)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">24</td><td>1+24=25</td></tr><tr><td align="center">2</td><td align="center">12</td><td>2+12=14</td></tr><tr><td align="center">3</td><td align="center">8</td><td>3+8=11</td></tr><tr><td align="center">4</td><td align="center">6</td><td>4+6=10</td></tr><tr><td align="center">-1</td><td align="center">-24</td><td>-1+(-24)=-25</td></tr><tr><td align="center">-2</td><td align="center">-12</td><td>-2+(-12)=-14</td></tr><tr><td align="center">-3</td><td align="center">-8</td><td>-3+(-8)=-11</td></tr><tr><td align="center">-4</td><td align="center">-6</td><td>-4+(-6)=-10</td></tr></table>



From this list we can see that 3 and 8 add up to 11 and multiply to 24



Now looking at the expression {{{2p^2+11p+12}}}, replace {{{11p}}} with {{{3p+8p}}} (notice {{{3p+8p}}} adds up to {{{11p}}}. So it is equivalent to {{{11p}}})


{{{2p^2+highlight(3p+8p)+12}}}



Now let's factor {{{2p^2+3p+8p+12}}} by grouping:



{{{(2p^2+3p)+(8p+12)}}} Group like terms



{{{p(2p+3)+4(2p+3)}}} Factor out the GCF of {{{p}}} out of the first group. Factor out the GCF of {{{4}}} out of the second group



{{{(p+4)(2p+3)}}} Since we have a common term of {{{2p+3}}}, we can combine like terms


So {{{2p^2+3p+8p+12}}} factors to {{{(p+4)(2p+3)}}}



So this also means that {{{2p^2+11p+12}}} factors to {{{(p+4)(2p+3)}}} (since {{{2p^2+11p+12}}} is equivalent to {{{2p^2+3p+8p+12}}})