Question

You synthetic division to find P(3) for p(x)=x^4-6x^3-4x^2-6x-2

Answer 1

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 $p(3)=-137$.