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