Use the factor theorem to determine whether $g\left(x\right)$ is a factor of $f\left(x\right)$:

(ii) $f\left(x\right)=2x^3+4x+6;g\left(x\right)=x+1$

Checking whether $g\left(x\right)$ is a factor of $f\left(x\right)$ or not.

Factor theorem: Let $f\left(x\right)$ be a polynomial of degree greater than or equal to 1 and $a$ is any real number, then

$f\left(a\right)=0\Rightarrow (x-a)$ is a factor of $f\left(x\right)$

$(x-a)$ is a factor of $f\left(x\right)$ $\Rightarrow$$f\left(a\right)=0$.

According to the factor theorem, $g\left(x\right)isafactoroff\left(x\right),iff(-1)=0[\sin ce,g(x)\Rightarrow x+1=0\Rightarrow x=-1]$

Substitute $-1$ for $x$in the function $f\left(x\right)$ and simplify.

$$\begin{gathered} f(-1)={2(-1)}^3+4(-1)+6 \\ \Rightarrow f(-1)=-2-4+6 \\ \therefore f(-1)=0 \end{gathered}$$

As $f(-1)=0$, the given polynomial $g\left(x\right)$ is a factor of $f\left(x\right)$.

Hence, the polynomial $g\left(x\right)=x+1$ is a factor of $f\left(x\right)=2x^3+4x+6$