Determine which of the following polynomials has $(x + 1)$ a factor : $x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2}$

For $x+1=0,$ we have $x=-1$.

$\therefore $ The zero of $x+1$ is $-1$.

$\because$ $p ( x )= x ^{3}- x ^{2}-(2+\sqrt{2}) x +\sqrt{2}$

$\therefore$ $p(-1)=(-1)^{3}-(-1)^{2}-(2+\sqrt{2})(-1)+\sqrt{2}$

$=-1-1-f(-1)(2+\sqrt{2})+\sqrt{2}=-1-1+1(2 \cdot+\sqrt{2})+\sqrt{2}$

$=-1-1 \cdot+2+\sqrt{2}+\sqrt{2}=-2+2+2 \sqrt{2}=2 \sqrt{2} \neq 0$

since $p (-1) \neq 0$.

$\therefore(x+1)$ is not a factor of $x^{4}+3 x^{3}+3 x^{2}+x+1$.