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t^6 - 7 t^3 - 8 \\ \\ (t^3 - 8)(t^3 + 1) \\ \\ (t^3 - 2^3)(t^3+1^3) \\ \\ (t -2)(t^2+2t+4)(t+1)(t^2-t+1) \\

for factorization some times, it is easier to find factors by looking at the coefficients. constant term divided by coefficient of highest power term.

- 8 / 1 : factors are :  +- 8 /1 , +- 4/1 , +- 2/1  +- 1/1 : 

some of these eight are possible values that appear as constants in factors. In this case the factors are : (x -8) (x +1)
actually , i finally asked my teacher and a simpler explanation was to take t cube as x
if u do that, you can factorize in the first step. good. it is correct. BUT after the first step, we have to do further factorization. like (t-2)(t^2+2t+4) ? terms with tcube are to be factorized also.
my explanation in the above answer like -8/1 factors are: that was additional information only. the basic working steps i showed in the first 4 lines of answer. thanks for feedback.