2018 AMC 12A Problem 5

Since $x^2 - 3x + 2 = (x-1)(x-2),$ its roots are $1$ and $2.$ If $1$ is a shared root then $1 - 5 + k = 0,$ so $k = 4.$ If $2$ is a shared root then $4 - 10 + k = 0,$ so $k = 6.$ The sum of possible values is $4 + 6 = 10.$ Thus, the correct answer is E.