Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$ which satisfy the division algorithm and $\operatorname{deg} q(x) = \operatorname{deg} r(x)$.

(N/A) According to the division algorithm,if $p(x)$ and $g(x)$ are two polynomials with $g(x) \neq 0$,then we can find polynomials $q(x)$ and $r(x)$ such that $p(x) = g(x) \cdot q(x) + r(x)$,where $r(x) = 0$ or $\operatorname{deg} r(x) < \operatorname{deg} g(x)$.
We need to find examples such that $\operatorname{deg} q(x) = \operatorname{deg} r(x)$.
Let us consider $p(x) = x^3 + x$ and $g(x) = x^2$.
Performing the division:
$x^3 + x = (x^2) \cdot x + x$.
Here,$q(x) = x$ and $r(x) = x$.
Checking the conditions:
$1$. $\operatorname{deg} q(x) = \operatorname{deg}(x) = 1$.
$2$. $\operatorname{deg} r(x) = \operatorname{deg}(x) = 1$.
Since $1 = 1$,the condition $\operatorname{deg} q(x) = \operatorname{deg} r(x)$ is satisfied.
$3$. $\operatorname{deg} r(x) < \operatorname{deg} g(x)$ is $1 < 2$,which is true.
Thus,the division algorithm is satisfied.