For this one, focus on the matched pair first:
4+sqrt(7), 4-sqrt(7)
I'll call these two roots R and Q
(x-R)(x-Q) = x^2 -Rx -Qx + RQ
The sqrt() parts cancel each other out in the x terms. And in the RQ term, you get 16-7 = 9.
so (x-R)(x-Q) = x^2 - 8x + 9
Then multiply this by the third root:
(x^2 - 8x + 9)(x-6) = x^3 -8x^2 + 9 x
-6x^2 +48 x -54
= x^3 -14 x^2 + 57 x - 54