Question 115655


Looking at {{{4x^2+4x-99}}} we can see that the first term is {{{4x^2}}} and the last term is {{{-99}}} where the coefficients are 4 and -99 respectively.


Now multiply the first coefficient 4 and the last coefficient -99 to get -396. Now what two numbers multiply to -396 and add to the  middle coefficient 4? Let's list all of the factors of -396:




Factors of -396:

1,2,3,4,6,9,11,12,18,22,33,36,44,66,99,132,198,396


-1,-2,-3,-4,-6,-9,-11,-12,-18,-22,-33,-36,-44,-66,-99,-132,-198,-396 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -396

(1)*(-396)

(2)*(-198)

(3)*(-132)

(4)*(-99)

(6)*(-66)

(9)*(-44)

(11)*(-36)

(12)*(-33)

(18)*(-22)

(-1)*(396)

(-2)*(198)

(-3)*(132)

(-4)*(99)

(-6)*(66)

(-9)*(44)

(-11)*(36)

(-12)*(33)

(-18)*(22)


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



Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-396</td><td>1+(-396)=-395</td></tr><tr><td align="center">2</td><td align="center">-198</td><td>2+(-198)=-196</td></tr><tr><td align="center">3</td><td align="center">-132</td><td>3+(-132)=-129</td></tr><tr><td align="center">4</td><td align="center">-99</td><td>4+(-99)=-95</td></tr><tr><td align="center">6</td><td align="center">-66</td><td>6+(-66)=-60</td></tr><tr><td align="center">9</td><td align="center">-44</td><td>9+(-44)=-35</td></tr><tr><td align="center">11</td><td align="center">-36</td><td>11+(-36)=-25</td></tr><tr><td align="center">12</td><td align="center">-33</td><td>12+(-33)=-21</td></tr><tr><td align="center">18</td><td align="center">-22</td><td>18+(-22)=-4</td></tr><tr><td align="center">-1</td><td align="center">396</td><td>-1+396=395</td></tr><tr><td align="center">-2</td><td align="center">198</td><td>-2+198=196</td></tr><tr><td align="center">-3</td><td align="center">132</td><td>-3+132=129</td></tr><tr><td align="center">-4</td><td align="center">99</td><td>-4+99=95</td></tr><tr><td align="center">-6</td><td align="center">66</td><td>-6+66=60</td></tr><tr><td align="center">-9</td><td align="center">44</td><td>-9+44=35</td></tr><tr><td align="center">-11</td><td align="center">36</td><td>-11+36=25</td></tr><tr><td align="center">-12</td><td align="center">33</td><td>-12+33=21</td></tr><tr><td align="center">-18</td><td align="center">22</td><td>-18+22=4</td></tr></table>



From this list we can see that -18 and 22 add up to 4 and multiply to -396



Now looking at the expression {{{4x^2+4x-99}}}, replace {{{4x}}} with {{{-18x+22x}}} (notice {{{-18x+22x}}} adds up to {{{4x}}}. So it is equivalent to {{{4x}}})


{{{4x^2+highlight(-18x+22x)+-99}}}



Now let's factor {{{4x^2-18x+22x-99}}} by grouping:



{{{(4x^2-18x)+(22x-99)}}} Group like terms



{{{2x(2x-9)+11(2x-9)}}} Factor out the GCF of {{{2x}}} out of the first group. Factor out the GCF of {{{11}}} out of the second group



{{{(2x+11)(2x-9)}}} Since we have a common term of {{{2x-9}}}, we can combine like terms


So {{{4x^2-18x+22x-99}}} factors to {{{(2x+11)(2x-9)}}}



So this also means that {{{4x^2+4x-99}}} factors to {{{(2x+11)(2x-9)}}} (since {{{4x^2+4x-99}}} is equivalent to {{{4x^2-18x+22x-99}}})


-------------------------------

Answer:


So {{{4x^2+4x-99}}} factors to {{{(2x+11)(2x-9)}}}