I am solving this question given in book (Automatic control system). As asked in (a) part $G_c(s)$ of the controller. I solved it and getting answer$$G_c(s) = \frac{F(s)}{E_c(s)}=\frac{100}{s}-\frac{30}{s+6}+\frac{70}{s+10}$$ but the answer in manual is $$G_c(s) = \frac{F(s)}{E_c(s)}=s(\frac{100}{s}-\frac{30}{s+6}+\frac{70}{s+10})$$ I know laplace transform and i am not too mature in control theory so please help in solving this question. I want to know from where this $s$ in multiplication is coming.
I don't think the issue is really about control theory. It is about notation.
The LT of the numerator, $F(s)$, is precisely the quantity in parentheses. That is including the step function factor $u_s(t)$. Thus, you cannot simply cancel the $u_s(t)$ from the numerator and denominator.
Thus, the LT of the denominator, $E_C(s)$, is the LT of the unit step function $u_s(t)$, or $1/s$. That accounts for the missing factor of $s$ in the answer.