Question 171351
I'm assuming that you want to factor





{{{75b^2-180b+108}}} Start with the given expression



{{{3(25b^2-60b+36)}}} Factor out the GCF {{{3}}}



Now let's focus on the inner expression {{{25b^2-60b+36}}}





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Looking at {{{25b^2-60b+36}}} we can see that the first term is {{{25b^2}}} and the last term is {{{36}}} where the coefficients are 25 and 36 respectively.


Now multiply the first coefficient 25 and the last coefficient 36 to get 900. Now what two numbers multiply to 900 and add to the  middle coefficient -60? Let's list all of the factors of 900:




Factors of 900:

1,2,3,4,5,6,9,10,12,15,18,20,25,30,36,45,50,60,75,90,100,150,180,225,300,450


-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-25,-30,-36,-45,-50,-60,-75,-90,-100,-150,-180,-225,-300,-450 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 900

1*900

2*450

3*300

4*225

5*180

6*150

9*100

10*90

12*75

15*60

18*50

20*45

25*36

30*30

(-1)*(-900)

(-2)*(-450)

(-3)*(-300)

(-4)*(-225)

(-5)*(-180)

(-6)*(-150)

(-9)*(-100)

(-10)*(-90)

(-12)*(-75)

(-15)*(-60)

(-18)*(-50)

(-20)*(-45)

(-25)*(-36)

(-30)*(-30)


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



Now which of these pairs add to -60? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -60


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">900</td><td>1+900=901</td></tr><tr><td align="center">2</td><td align="center">450</td><td>2+450=452</td></tr><tr><td align="center">3</td><td align="center">300</td><td>3+300=303</td></tr><tr><td align="center">4</td><td align="center">225</td><td>4+225=229</td></tr><tr><td align="center">5</td><td align="center">180</td><td>5+180=185</td></tr><tr><td align="center">6</td><td align="center">150</td><td>6+150=156</td></tr><tr><td align="center">9</td><td align="center">100</td><td>9+100=109</td></tr><tr><td align="center">10</td><td align="center">90</td><td>10+90=100</td></tr><tr><td align="center">12</td><td align="center">75</td><td>12+75=87</td></tr><tr><td align="center">15</td><td align="center">60</td><td>15+60=75</td></tr><tr><td align="center">18</td><td align="center">50</td><td>18+50=68</td></tr><tr><td align="center">20</td><td align="center">45</td><td>20+45=65</td></tr><tr><td align="center">25</td><td align="center">36</td><td>25+36=61</td></tr><tr><td align="center">30</td><td align="center">30</td><td>30+30=60</td></tr><tr><td align="center">-1</td><td align="center">-900</td><td>-1+(-900)=-901</td></tr><tr><td align="center">-2</td><td align="center">-450</td><td>-2+(-450)=-452</td></tr><tr><td align="center">-3</td><td align="center">-300</td><td>-3+(-300)=-303</td></tr><tr><td align="center">-4</td><td align="center">-225</td><td>-4+(-225)=-229</td></tr><tr><td align="center">-5</td><td align="center">-180</td><td>-5+(-180)=-185</td></tr><tr><td align="center">-6</td><td align="center">-150</td><td>-6+(-150)=-156</td></tr><tr><td align="center">-9</td><td align="center">-100</td><td>-9+(-100)=-109</td></tr><tr><td align="center">-10</td><td align="center">-90</td><td>-10+(-90)=-100</td></tr><tr><td align="center">-12</td><td align="center">-75</td><td>-12+(-75)=-87</td></tr><tr><td align="center">-15</td><td align="center">-60</td><td>-15+(-60)=-75</td></tr><tr><td align="center">-18</td><td align="center">-50</td><td>-18+(-50)=-68</td></tr><tr><td align="center">-20</td><td align="center">-45</td><td>-20+(-45)=-65</td></tr><tr><td align="center">-25</td><td align="center">-36</td><td>-25+(-36)=-61</td></tr><tr><td align="center">-30</td><td align="center">-30</td><td>-30+(-30)=-60</td></tr></table>



From this list we can see that -30 and -30 add up to -60 and multiply to 900



Now looking at the expression {{{25b^2-60b+36}}}, replace {{{-60b}}} with {{{-30b+-30b}}} (notice {{{-30b+-30b}}} adds up to {{{-60b}}}. So it is equivalent to {{{-60b}}})


{{{25b^2+highlight(-30b+-30b)+36}}}



Now let's factor {{{25b^2-30b-30b+36}}} by grouping:



{{{(25b^2-30b)+(-30b+36)}}} Group like terms



{{{5b(5b-6)-6(5b-6)}}} Factor out the GCF of {{{5b}}} out of the first group. Factor out the GCF of {{{-6}}} out of the second group



{{{(5b-6)(5b-6)}}} Since we have a common term of {{{5b-6}}}, we can combine like terms


So {{{25b^2-30b-30b+36}}} factors to {{{(5b-6)(5b-6)}}}



So this also means that {{{25b^2-60b+36}}} factors to {{{(5b-6)(5b-6)}}} (since {{{25b^2-60b+36}}} is equivalent to {{{25b^2-30b-30b+36}}})



note:  {{{(5b-6)(5b-6)}}} is equivalent to  {{{(5b-6)^2}}} since the term {{{5b-6}}} occurs twice. So {{{25b^2-60b+36}}} also factors to {{{(5b-6)^2}}}




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So our expression goes from {{{3(25b^2-60b+36)}}} and factors further to {{{3(5b-6)^2}}}



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Answer:


So {{{75b^2-180b+108}}} factors to {{{3(5b-6)^2}}}