Solving a block diagram problem of an LTI-System
I'd like to show you a general way to solve such problems. It's always helpful to define auxiliary variables at the outputs of the adders. You have three adders, but the output of the adder in the middle is the system's output \$Y(s)\$, so you need only two auxiliary variables. \$A_1(s)\$ is the output of the left-most adder, and \$A_2(s)\$ is the output of the bottom adder. For each adder you get one equation (for the sake of convenience and readability, I leave out the argument \$s\$):

$$\begin{align}A_1&amp;=X+H_1Y\\
Y&amp;=H_2A_1-A_2+Y\\
A_2&amp;=H_3A_1+H_4Y\end{align}$$

It is straightforward to solve these equations for the unknown total transfer function

$$H(s)=\frac{Y(s)}{X(s)}=\frac{H_2(s)-H_3(s)}{H_4(s)-H_1(s)[H_2(s)-H_3(s)]}$$

Note that this result is identical to the one presented in hypfco's answer.