Find the value of $k,$ if $x-1$ is a factor of $4x^{3}+3x^{2}-4x+k$.

What are the possible expressions for the dimensions of a cuboid whose volume is given as $(2x^3 + 15x^2 + 33x + 20)$ cubic units? $(x > 0)$

Determine which of the following polynomials has $(x + 1)$ as a factor: $x^{4} + 3x^{3} + 3x^{2} + x + 1$.

Factorise: $2y^{3} + y^{2} - 2y - 1$

Find the remainder obtained on dividing $p(x) = x^3 + 1$ by $x + 1$.

Without actual division,prove that $2x^4 - 5x^3 + 2x^2 - x + 2$ is divisible by $x^2 - 3x + 2$.