$a)$

$\left(u-w\right)\left(x\right)=u\left(x\right)-w\left(x\right)=a{x}^4+x^2-1-b{x}^3+3x+2=a{x}^4-b{x}^3+x^2+3x+1$

$\{\begin{array}{l}a\ne 0;b\in \mathbb{R}\Rightarrow st\left(u-w\right)=4\\ a=0;b\ne 0\Rightarrow st\left(u-w\right)=3\\ a=0;b=0\Rightarrow st\left(u-w\right)=2\end{array}$

 

 

 

$b)$

$\left(u-w\right)\left(x\right)=a{x}^5+x^4+3x-2b{x}^4-2x+x^5+5=\left(a+1\right)x^5+\left(1-2b\right)x^4+x+5$

$\{\begin{array}{l}a+1\ne 0;b\in \mathbb{R}\Rightarrow a\in \mathbb{R}\backslash \left\{-1\right\};b\in \mathbb{R}\Rightarrow st\left(u-w\right)=5\\ a+1=0;1-2b\ne 0\Rightarrow a=-1;b\in \mathbb{R}\backslash \left\{\frac{1}{2}\right\}\Rightarrow st\left(u-w\right)=4\\ a=-1;b=\frac{1}{2}\Rightarrow st\left(u-w\right)=1\end{array}$

 

   

 

$c)$

$\left(u-w\right)\left(x\right)=2a{x}^4+x^2+3+2b{x}^4+a{x}^3+7x^2=\left(2a+2b\right)x^4+a{x}^3+8x^2+3$

$\{\begin{array}{l}2a+2b\ne 0\Rightarrow 2a\ne -2b\Rightarrow a\ne -b\Rightarrow st\left(u-w\right)=4\\ 2a+2b\ne 0;a\ne 0\Rightarrow a=-b\ne 0\Rightarrow st\left(u-w\right)=3\\ 2a+2b=0;a=0\Rightarrow a=b=0\Rightarrow st\left(u-w\right)=2\end{array}$