The polynomial equation of degree $4$ having real coefficients with three of its roots as $2 \pm \sqrt{3}$ and $1+2i$ is:

What is the remainder of $2x^3 - 5x^2 + 7$ when divided by $(x - 2)$?

If $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ $(a \neq 0)$ and $\alpha+h, \beta+h$ are the roots of $px^{2}+qx+r=0$ $(p \neq 0),$ then the ratio of the squares of their discriminants is

Let $\alpha, \beta, \gamma, \delta$ be the roots of the equation $x^4 + x^2 + 1 = 0$. Then the equation whose roots are $\alpha^2, \beta^2, \gamma^2, \delta^2$ is:

If for some $p, q, r \in \mathbb{R}$,not all have the same sign,and one of the roots of the equation $(p^{2}+q^{2})x^{2}-2q(p+r)x + q^{2}+r^{2}=0$ is also a root of the equation $x^{2}+2x-8=0$,then $\frac{q^{2}+r^{2}}{p^{2}}$ is equal to-

If $2+\sqrt{3}$ is a root of the equation $f(x)=x^4+2x^3-16x^2-22x+7=0$,then which one of the following is not a root of $f(x)=0$?