The clue to the solution lies in the denominator, because the factors are recognisable as denominators of Laplace Transforms in a table. So the first step is to convert the expression into partial fractions.
F(s)=A/(s-1)+B/(s-2)+C/(s-4) where A, B and C are constants.
If the transform is 1/(s-a), then the inverse is eᵃᵗ, so f(t)=Aeᵗ+Be²ᵗ+Ce⁴ᵗ.
To find A, B and C we use the identity:
A+B+C=1, 6A+5B+3C=-6, 8A+4B+2C=9.
There are 3 equations and 3 unknowns in this system so we can find the unknowns.
3A=7-(-9)=16, A=16/3; 2B=7-6A=7-32=-25, B=-25/2;
16/3-25/2+C=1, 32/6-75/6+C=1, C=75/6-32/6+6/6=49/6.