We are given the determinant:

|(a, 3, 4), (1, 2, 1), (1, 4, 1)| = 0

Expanding the determinant along the first row, we get:

a * |(2, 1), (4, 1)| - 3 * |(1, 1), (1, 1)| + 4 * |(1, 2), (1, 4)| = 0

Calculating the 2x2 determinants:

a * (2*1 - 1*4) - 3 * (1*1 - 1*1) + 4 * (1*4 - 2*1) = 0

a * (2 - 4) - 3 * (1 - 1) + 4 * (4 - 2) = 0

a * (-2) - 3 * (0) + 4 * (2) = 0

-2a + 0 + 8 = 0

-2a + 8 = 0

Solving for a:

-2a = -8

a = -8 / -2

a = 4