The first thing to try is to see if there are any common factors you can cancel out.
color(blue)("Consider the numerator")Consider the numerator
3 is prime so I can not factor out any constants from all of the numerator
color(blue)("Try 1:") ->(3t-1)(t-4) = 3t^2-12t-4t+4 color(red)(larr" Fail")Try 1:→(3t−1)(t−4)=3t2−12t−4t+4← Fail
color(blue)("Try 2:") " Write as: "color(green)( 3t^2-6t-2t+4)color(purple)( -> 3t(t-2)-2(t-2))Try 2: Write as: 3t2−6t−2t+4→3t(t−2)−2(t−2)
Giving: " "(t-2)(3t-2)color(red)(" "larr" Works") (t−2)(3t−2) ← Works
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Consider the denominator")Consider the denominator
Factor out 2t" giving "2t(3t-2)2t giving 2t(3t−2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Putting it all together")Putting it all together
(3t^2-8t_4)/(6t^2-4t) -= ((t-2)cancel((3t-2)))/(2tcancel((3t-2)))
giving:" "1/(2t)(t-2) = 1/2-1/t
