Give an example of polynomials f(x), g(x),

Using division algorithm, we have

$f(x)=g(x) \times q(x)+r(x)$

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

$x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-3 x^{3}+x^{2}-x^{3}+3 x-1+2$

$x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-3 x^{3}-x^{3}+x^{2}+3 x-1+2$

$x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-4 x^{3}+x^{2}+3 x+1$

Hence an example for polynomial $f(x), g(x), q(x)$ and $r(x)$ satisfying $f(x)=g(x) \times q(x)+r(x)$ are

$f(x)=x^{5}-4 x^{3}+x^{2}+3 x+1$

$g(x)=\left(x^{3}-3 x+1\right)$

$q(x)=\left(x^{2}-1\right)$

$r(x)=2$