The angles $\theta$ and $\varphi$ lie between $0°$ and $180°,$ and are such that $\tan (\theta -\varphi )=3$ and $\tan \theta +\tan \varphi =1.$ Find the possible values of $\theta$ and $\varphi .$

The polynomial $4x^3+a{x}^2+9x+9,$ where $a$ is a constant, is denoted by $\mathrm{p}\left(\mathrm{x}\right).$ It is given that when $\mathrm{p}\left(\mathrm{x}\right)$ is divided by $(2x-1)$ the remainder is $10.$ Find the value of $a$ and hence verify that $\left(x-3\right)$ is a factor of $\mathrm{p}\left(\mathrm{x}\right).$

The polynomial $4x^3+a{x}^2+9x+9,$ where $a$ is a constant, is denoted by $\mathrm{p}\left(\mathrm{x}\right).$ It is given that when $\mathrm{p}\left(\mathrm{x}\right)$ is divided by $(2x-1)$ the remainder is $10.$ When $a=-16$, solve the equation $\mathrm{p}\left(\mathrm{x}\right)=0.$.

The polynomial $2x^3-4x^2+ax+b,$ where $a$ and $b$ are constants, is denoted by $\mathrm{p}\left(x\right).$ It is given that when $\mathrm{p}\left(x\right)$ is divided by $(x+1)$ the remainder is $4,$ and that when $\mathrm{p}\left(x\right)$ is divided by $\left(x-3\right)$ the remainder is $12.$ Find the values of $a$ and $b.$

The polynomial $2x^3-4x^2+ax+b,$ where $a$ and $b$ are constants, is denoted by $\mathrm{p}\left(x\right).$ It is given that when $\mathrm{p}\left(x\right)$ is divided by $(x+1)$ the remainder is $4,$ and that when $\mathrm{p}\left(x\right)$ is divided by $\left(x-3\right)$ the remainder is $12.$ When $a=-4$ and $b=6$, find the quotient and remainder when $p\left(x\right)$ is divided by $\left(x^2-2\right).$