$a,b\in R$,roots of $x^2+ax+b=0$ are $\alpha,\beta$ and roots of $x^2+bx+a=0$ are $\alpha+k,\beta+k$ where $k\neq 0$.Which of the following is true?

Question

$\begin{array}{1 1}(A)\;a+b+1=0&(B)\;a=b+1\\(C)\;a+b+4=0&(D)\;\text{None}\end{array}$