Nice answer. One thing might be mentioned that if matrix $A$ has Eigen value $\lambda$ and $B$ has Eigen value $\mu$ then matrix $AB$ has Eigen value $\lambda \mu$ if matrices $A,B$ and $AB$ share the same set of Eigen vectors..It can be proved easily...Here, matrices $(x+I)^{-1}$ and $(x+5I)$ always have the same set of Eigen vectors. Matrices $x,(x+I),(x+I)^{-1},(x+5I),(x+I)^{-1}(x+5I)$ all have the same set of Eigen vectors..This might be important because for similar kind of questions, there may give another matrix $B$ with Eigen value $\lambda$ and ask about the Eigen value $(x+I)^{-1}B$.

@ankitgupta.1729yes my doubt is there only..I am asking him that only..same goes for addition also..that they must share common eigen vectors..then only we can add their eigen value