Question 139589
{{{8+45r-18r^2}}} Start with the given expression



{{{-18r^2+45r+8}}} Sort the terms in descending order



{{{-(18r^2-45r-8)}}} Factor out a negative one



Looking at the inner polynomial {{{18r^2-45r-8}}}, we can see that the first term is {{{18r^2}}} and the last term is -8


Now multiply the first coefficient 18 and the last term -8 to get -144. Now what two numbers multiply to -144 and add to the  middle coefficient -45? Let's list all of the factors of -144:




Factors of -144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72


-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -144

(1)*(-144)

(2)*(-72)

(3)*(-48)

(4)*(-36)

(6)*(-24)

(8)*(-18)

(9)*(-16)

(-1)*(144)

(-2)*(72)

(-3)*(48)

(-4)*(36)

(-6)*(24)

(-8)*(18)

(-9)*(16)


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



Now which of these pairs add to -45? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -45


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-144</td><td>1+(-144)=-143</td></tr><tr><td align="center">2</td><td align="center">-72</td><td>2+(-72)=-70</td></tr><tr><td align="center">3</td><td align="center">-48</td><td>3+(-48)=-45</td></tr><tr><td align="center">4</td><td align="center">-36</td><td>4+(-36)=-32</td></tr><tr><td align="center">6</td><td align="center">-24</td><td>6+(-24)=-18</td></tr><tr><td align="center">8</td><td align="center">-18</td><td>8+(-18)=-10</td></tr><tr><td align="center">9</td><td align="center">-16</td><td>9+(-16)=-7</td></tr><tr><td align="center">-1</td><td align="center">144</td><td>-1+144=143</td></tr><tr><td align="center">-2</td><td align="center">72</td><td>-2+72=70</td></tr><tr><td align="center">-3</td><td align="center">48</td><td>-3+48=45</td></tr><tr><td align="center">-4</td><td align="center">36</td><td>-4+36=32</td></tr><tr><td align="center">-6</td><td align="center">24</td><td>-6+24=18</td></tr><tr><td align="center">-8</td><td align="center">18</td><td>-8+18=10</td></tr><tr><td align="center">-9</td><td align="center">16</td><td>-9+16=7</td></tr></table>



From this list we can see that 3 and -48 add up to -45 and multiply to -144



Now looking at the expression {{{18r^2-45r-8}}}, replace {{{-45r}}} with {{{3r+-48r}}} (notice {{{3r+-48r}}} adds up to {{{-45r}}}. So it is equivalent to {{{-45r}}})


{{{18r^2+highlight(3r+-48r)+-8}}}



Now let's factor {{{18r^2+3r-48r-8}}} by grouping:



{{{(18r^2+3r)+(-48r-8)}}} Group like terms



{{{3r(6r+1)-8(6r+1)}}} Factor out the GCF of {{{3r}}} out of the first group. Factor out the GCF of {{{-8}}} out of the second group



{{{(3r-8)(6r+1)}}} Since we have a common term of {{{6r+1}}}, we can combine like terms


So {{{18r^2+3r-48r-8}}} factors to {{{(3r-8)(6r+1)}}}



So this also means that {{{18r^2-45r-8}}} factors to {{{(3r-8)(6r+1)}}} (since {{{18r^2-45r-8}}} is equivalent to {{{18r^2+3r-48r-8}}})




So {{{-(18r^2-45r-8)}}} factors to {{{-(3r-8)(6r+1)}}}


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Answer:


So {{{8+45r-18r^2}}} factors to {{{-(3r-8)(6r+1)}}}