Find the remainder when $x^3+3x^2+3x+1$ is divided by $x+\mathrm{\pi }.$

Determine if the following polynomial has $\left(x+1\right)$ a factor:

$x^3+x^2+x+1$

Determine if the following polynomial has $\left(x+1\right)$ a factor:

$x^4+x^3+x^2+x+1$

Determine if the following polynomial has $\left(x+1\right)$ a factor:

$x^4+3x^3+3x^2+x+1$

Determine if the following polynomial has $\left(x+1\right)$ a factor:

$x^3-x^2-\left(2+\sqrt{2}\right)x+\sqrt{2}$

Use the Factor Theorem to determine whether $g\left(x\right)$ is a factor of $p\left(x\right)$ in the following case:

$p\left(x\right)=2x^3+x^2-2x-1, g\left(x\right)=x+1$

Use the Factor Theorem to determine whether $g\left(x\right)$ is a factor of $p\left(x\right)$ in the following case:

$p\left(x\right)=x^3-4x^2+x+6, g\left(x\right)=x-3$

Factorise: $x^3-3x^2-9x-5$