We can guess what the zeroes are by looking at the constant term 21 which has rational factors 3 and 7, as well as 1 and 21. Possible rational factors are also the negatives of these values.
Always try x=1 and x=-1 as possible factors x-1 and x+1. Simply write out the polynomial and note the coefficients: x=1 gives us 76-76=0, so x-1 is a factor:
Using synthetic division for x=1:
1|6 35 35 -55 -21
6 6 41 76 | 21
6 41 76 21 | 0 = 6x3+41x2+76x+21.
The polynomial is the same as: (x-1)(6x3+41x2+76x+21).
We can see that x=1 or x=-1 are not factors of 6x3+41x2+76x+21.
All the coefficients are positive, so we are looking for a negative zero which is a factor of 21. The smallest is -3, so let's see if it's a factor:
-3| 6 41 76 21
6 -18 -69 | -21
6 23 7 | 0 = 6x2+23x+7=(2x+7)(3x+1).
Not only can we see that x+3 is a factor because the remainder is zero, but the quadratic quotient factorises, so we have all four zeroes, the remaining two being -7/2 and -⅓.
Complete factorisation is (x-1)(x+3)(2x+7)(3x+1).