Find the possible value of k if X^2+(k-2)X+K+1=0 has equal roots

if the roots are equal, then the discriminant is zero. So,

(k-2)^2 - 4(1)(k+1) = 0
k^2 - 4k + 4 - 4k - 4 = 0
k^2 - 8k = 0
k = 0 or 8

you really should check to make sure that those two values of k cause the polynomial to have equal roots. That is, it will be a perfect square.