If $x=2$ is a root of the polynomial $f\left(x\right)=2x^2-3x+7a,$ then find the value of $a$.

In each of the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division:

$f\left(x\right)=x^3+4x^2-3x+10, g\left(x\right)=x+4$

In the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division:

In the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division:

In the following, using the remainder theorem, find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$ and verify the result by actual division: