What is the value of k for which (k² – 5k + 4)x² + (k² – 3k – 4)x + (k² – 4k) = 0 is an identity?

For the equation to be an identity (true for all x), every coefficient must equal zero simultaneously. Setting k² – 5k + 4 = 0 gives k = 1 or k = 4. Setting k² – 3k – 4 = 0 gives k = 4 or k = –1. Setting k² – 4k = 0 gives k = 0 or k = 4. The common value satisfying all three equations is k = 4.