Let λ be an eigenvalue of A and X be the eigenvector of A.

Then according to the definition of eigenvalues and eigenvector we get

AX = λ X ........................(1)

Now, multiplying both sides of equation 1 with A we get

A^{2}X = A λ X = λ AX = λ λ X = λ^{2}X ............... (2)

So, the matrix A^{2 }has the same eigenvector as in A but the eigenvalue is square of the eigenvalue of A.

Now, multiplying both sides of equation 1 with -3 we get

-3 AX = -3 λX ..........................(3)

Now, adding equation (2) and (3) we get

(A^{2} - 3A)X = ( λ^{2} - 3 λ )X ........................(4)

Now, we add 4$I$ X to the both sides of equation 4

(A^{2} - 3A) X + 4$I$ X = ( λ^{2} - 3 λ )X + 4$I$X

=> (A^{2} - 3A + 4$I$)X = ( λ^{2} - 3 λ +4 )X [ since $I$X = X]

So, the eigenvalue of the matrix (A^{2} - 3A +4$I$) is ( λ^{2} - 3 λ + 4 ) and the eigenvector remains same as A.

Now, the value of λ is -2 and 1

So, putting those values in the expression of the eigenvalue we get the eigenvalues of the given matrix which are 2 and 14

So, Option A is the correct answer.