Is (a-3) a factor of (a^3-6a^2+29)?

If (a - 3) is  a factor then (a -3) will be 0 when a = 3. If you put  a = 3 into the given cubic, you should get 0. Doing that, does not give 0.

f(3) = 8 - 6*9 + 29
f(3) = 8 - 54 + 29
f(3) = - 17

As further "proof", here is the graph. Notice that the graph cuts the x axis around -2 , 2.5 and 4.2 (These are rough guesses). The factors are
(a+2)(a - 2.5)(a - 4.2)