We have linear operator $T:C[0,1]\to[0,1]$ $$ Tf(t)=f(0)e^t+1/2f(1)(t+1) $$ Let $K=\text{lin}(e^t,t+1)$. We know that $\text{Im}T=K$. So we can easily find its eigenvalues which are $1-\sqrt{\frac{e}{2}}$ and $1+\sqrt{\frac{e}{2}}$.
Using this fact find spectrum of linear operator $T-I$, where $I$ is identity on $C[0,1]$.
Since the image of $T$ is finite dimensional it is a compact operator. For a compact operator the spectrum consists of the eigen values together with, possibly $0$. Obviously, $T$ is not invertible, so $0$ does belong to the spectrum. Hence $\sigma (T)=\{0,1-\sqrt {\frac e 2},1-\sqrt {\frac e 2}\}$. Ans $\sigma (T-I)=\{-1,-\sqrt {\frac e 2},-\sqrt {\frac e 2}\}$