So this is a solution to this partial fraction. I understand everything except the final line. I know how the first term 4x0.5/.... came about. I do NOT know how the second and third term 1/s - (....) came about from the previous line's second term 0.5/s[...]. Anyone care to show me how this is done pls. Thanks.
Hint $\ \dfrac{1}{x\,f(x)}\, =\, \dfrac{a}x -\dfrac{g(x)}{f(x)}\, \overset{\large {\rm times}\ x}\Rightarrow \dfrac{1}{f(x)}\, =\, a - \dfrac{x g(x)}{f(x)}\, \overset{\large x\,=\,0}\Rightarrow \dfrac{1}{f(0)}\, =\, a,\ \ $ if $\ \ f(0)\ne 0.$
Thus $\ \ \dfrac{g(x)}{f(x)}\, =\ \dfrac{1}{f(0)x} - \dfrac{1}{x\,f(x)}\,=\, \dfrac{\dfrac{f(x)-f(0)}{f(0)x}}{f(x)} $
In OP $\,f(x) = 2x^2\!+2x+1\,$ so above $\, = \dfrac{2x+2}{2x^2+2x+1} =\, \dfrac{(x+1/2)\,\ \,+\ 1/2\,\ \ \ }{(x+1/2)^2+(1/2)^2}$
