If $kx^3 + 4x^2 + 3x - 4 $ and $x^3 - 4x + k$ leave the same remainder when divided by (x - 3), then the value of k is:

-1

x - 3 = 0 

Then, x = 3

By substituting the value of x, we get

kx³ + 4x² + 3x - 4

= k(3)³ + 4(3)² + 3(3) - 4 = 27k + 36 + 9 - 4 = 27k + 41      ----(a)

x³ - 4x + k = 3³ - 4(3) + k = 27 - 12 + k = 15 + k      ----(b)

Both the dividends leave equal reminders  then equating (a) and (b)

27k + 41 = 15 + k

= 26k = -26

= k = -1