​ What are these two different methods?

One method

If x + 4 is a factor then - 4 is a root

So

f(-4) should = 0

So we have    4(-4)^4 - 2(-4)^3  -52(-4)^2  - 15 (-4) + 8 =

1024  + 128 - 832 + 60 + 8  =  388...so .... not a root

Another method....synthetic division

-4    [  4     -2     - 52     - 15     8   ]

                -16      72      -80  380

       ________________________

          4     -18    20      -95   388            same remainder as above  !!!

If it was a root....we sould get 0  as a result