Apply the division algorithm to find the quotient $q\left(x\right)$ and remainder $r\left(x\right)$ on dividing $p\left(x\right)=10x^4+8x^2+6x+3$ by $g\left(x\right)=x^2-5x+6$.

Check whether the polynomial $\mathrm{g}\left(\mathrm{x}\right)$ is a factor of the polynomial $\mathrm{p}\left(\mathrm{x}\right)$ by division algorithm, in the following cases.

$\mathrm{p}\left(\mathrm{x}\right)=2{\mathrm{x}}^4-3{\mathrm{x}}^3+4{\mathrm{x}}^2-7\mathrm{x}+4; \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2-2\mathrm{x}+1$

Check whether the polynomial $\mathrm{g}\left(\mathrm{x}\right)$ is a factor of the polynomial $\mathrm{p}\left(\mathrm{x}\right)$ by division algorithm, in the following cases.

$\mathrm{p}\left(\mathrm{x}\right)=6{\mathrm{x}}^4+{\mathrm{x}}^3+10{\mathrm{x}}^2+3\mathrm{x}+4; \mathrm{g}\left(\mathrm{x}\right)=2{\mathrm{x}}^2+\mathrm{x}+1$

Check whether the polynomial $\mathrm{g}\left(\mathrm{x}\right)$ is a factor of the polynomial $\mathrm{p}\left(\mathrm{x}\right)$ by division algorithm, in the following cases.

$\mathrm{p}\left(\mathrm{x}\right)=9{\mathrm{x}}^4+4{\mathrm{x}}^3-2{\mathrm{x}}^2+\mathrm{x}+10; \mathrm{g}\left(\mathrm{x}\right)=3{\mathrm{x}}^2+2\mathrm{x}+5$

Check whether the polynomial $\mathrm{g}\left(\mathrm{x}\right)$ is a factor of the polynomial $\mathrm{p}\left(\mathrm{x}\right)$ by division algorithm, in the following cases.

$\mathrm{p}\left(\mathrm{x}\right)=8{\mathrm{x}}^4+6{\mathrm{x}}^3-3{\mathrm{x}}^2+\mathrm{x}+6; \mathrm{g}\left(\mathrm{x}\right)=4{\mathrm{x}}^2+2\mathrm{x}+3$

Obtain all zeroes of $p\left(x\right)=x^4-3x^3-x^2+9x-6$ if two of its zeroes are $(-\sqrt{3})$ and $\sqrt{3}$.