What must be added to the polynomial $p(x) = 6x^5 + 5x^4 + 11x^3 - 3x^2 + x + 1$ so that the resulting polynomial is exactly divisible by $3x^2 - 2x + 4$? (Hint: add $-r(x)$ to $p(x)$)

Divide $3x^{3}+x^{2}+2x+5$ by $1+2x+x^{2}$.

The degree of the polynomial $p(x) = x^2 - x^3 + x + 1$ is $\ldots \ldots \ldots . .$

If the polynomial $p(x) = x^{3} - 3x^{2} + x + 2$ is divided by the divisor polynomial $s(x)$, then the quotient polynomial $q(x) = x - 2$ and the remainder polynomial $r(x) = -2x + 4$ are obtained. Find the divisor polynomial $s(x)$.

Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$ which satisfy the division algorithm and $\operatorname{deg} r(x) = 0$.

Divide $3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ by $x^{2}+3 x+1$.