First, we need to understand:
#t^2 - 9 = (t + 3)(t - 3)#
Therefore, we can multiply the two fractions on the left by the appropriate form of #1# to put them over a common denominator with the fraction on the right:
#((t - 3)/(t - 3) * (t + 1)/(t + 3)) - ((t + 3)/(t + 3) * (t - 2)/(t - 3)) + 6/(t^2 - 9) =>#
#((t - 3)(t + 1))/(t^2 - 9) - ((t + 3)(t - 2))/(t^2 - 9) + 6/(t^2 - 9) =>#
#(t^2 - 3t + 1t - 3)/(t^2 - 9) - (t^2 + 3t - 2t - 6)/(t^2 - 9) + 6/(t^2 - 9) =>#
#(t^2 + (-3 + 1)t - 3)/(t^2 - 9) - (t^2 + (3 - 2)t - 6)/(t^2 - 9) + 6/(t^2 - 9) =>#
#(t^2 + (-3)t - 3)/(t^2 - 9) - (t^2 + 1t - 6)/(t^2 - 9) + 6/(t^2 - 9) =>#
#(t^2 - 3t - 3)/(t^2 - 9) - (t^2 + 1t - 6)/(t^2 - 9) + 6/(t^2 - 9)#
Now, we can combine like terms in the numerator over the common denominator:
#(t^2 - 3t - 3 - t^2 - 1t + 6 + 6)/(t^2 - 9) =>#
#(t^2 - t^2 - 3t - 1t - 3 + 6 + 6)/(t^2 - 9) =>#
#((t^2 - t^2) + (-3 - 1)t + (-3 + 6 + 6))/(t^2 - 9) =>#
#(0 + (-4)t + 9)/(t^2 - 9) =>#
#(-4t + 9)/(t^2 - 9)#
Or
#(9 - 4t)/(t^2 - 9)#
