Question 148494
I'll do the first one to get you started.


a)




Looking at {{{1x^2-2x-24}}} we can see that the first term is {{{1x^2}}} and the last term is {{{-24}}} where the coefficients are 1 and -24 respectively.


Now multiply the first coefficient 1 and the last coefficient -24 to get -24. Now what two numbers multiply to -24 and add to the  middle coefficient -2? 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, the product of a negative and a positive number is a negative number



Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2


<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)=-23</td></tr><tr><td align="center">2</td><td align="center">-12</td><td>2+(-12)=-10</td></tr><tr><td align="center">3</td><td align="center">-8</td><td>3+(-8)=-5</td></tr><tr><td align="center">4</td><td align="center">-6</td><td>4+(-6)=-2</td></tr><tr><td align="center">-1</td><td align="center">24</td><td>-1+24=23</td></tr><tr><td align="center">-2</td><td align="center">12</td><td>-2+12=10</td></tr><tr><td align="center">-3</td><td align="center">8</td><td>-3+8=5</td></tr><tr><td align="center">-4</td><td align="center">6</td><td>-4+6=2</td></tr></table>



From this list we can see that 4 and -6 add up to -2 and multiply to -24



Now looking at the expression {{{1x^2-2x-24}}}, replace {{{-2x}}} with {{{4x+-6x}}} (notice {{{4x+-6x}}} adds up to {{{-2x}}}. So it is equivalent to {{{-2x}}})


{{{1x^2+highlight(4x+-6x)+-24}}}



Now let's factor {{{1x^2+4x-6x-24}}} by grouping:



{{{(1x^2+4x)+(-6x-24)}}} Group like terms



{{{x(x+4)-6(x+4)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-6}}} out of the second group



{{{(x-6)(x+4)}}} Since we have a common term of {{{x+4}}}, we can combine like terms


So {{{1x^2+4x-6x-24}}} factors to {{{(x-6)(x+4)}}}



So this also means that {{{1x^2-2x-24}}} factors to {{{(x-6)(x+4)}}} (since {{{1x^2-2x-24}}} is equivalent to {{{1x^2+4x-6x-24}}})




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     Answer:

So {{{x^2-2x-24}}} factors to {{{(x-6)(x+4)}}}