Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4) are coplanar.

For the four points A, B, C, D to be coplanar , they should lie on same plane which implies
AD · (AC × AB) = 0
AB = 4i + 6j + 2k, AC = i - 4j - 3k and AD = 8i + j - 3k
AB × AC = i(-18 + 8) + j(2 + 12) + k(-16 - 6) = -10i + 14j - 22k
AD · (AB × AC) = (8i + j - 3k) · (-10i + 14j - 22k) = -80 + 14 + 66 = 0
Therefore points A, B, C, D are coplanar.