If $-1$ is a twice repeated root of the equation $a(x^3+x^2)+bx+c=0$,then $a:b:c=$

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 ${b_1}{b_2} = 2({c_1} + {c_2})$,then at least one of the equations ${x^2} + {b_1}x + {c_1} = 0$ and ${x^2} + {b_2}x + {c_2} = 0$ has

If a polynomial $x^4+x^2+1$ is divisible by $x^2+mx+1$ and $x^2+nx+1$,then $m+n$ is equal to:
$(1)$ $2$
$(2)$ $0$
$(3)$ $3$
$(4)$ $4$

If $a + b + c = 0$,$a \ne 0$,and $a, b, c \in \mathbb{Q}$,then both the roots of the equation $ax^2 + bx + c = 0$ are

If $a > 0,$ then $\sqrt {a + \sqrt {a + \sqrt {a + ....\infty } } } $ is