By the polynomial remainder theorem, the remainder upon dividing \(p(x)\) by \(x-3\) will be the value of \(p(3)\).
... | 1 ... -6 ... -4 ... -6 .... -2
3. | .. ... 3 ... -9 ... -39 .. -135
--------------------------------------
... | 1 ... -3 ... -13 . -45 .. -137
So you have
\(\dfrac{x^4-6x^3-4x^2-6x-2}{x-3}=x^3-3x^2-13x-45-\dfrac{137}{x-3}\)
which means \(p(3)=-137\).
