I don't know how to solve this!

1. Find all possibles values of k such that the equation $\displaystyle x^2-(k-3)x+k^2+2k+5=0$ has real roots. If $\displaystyle \alpha$ and $\displaystyle \beta$ are real roots of this equation, show that $\displaystyle \alpha^2+\beta^2=-(k+5)^2+24$. Hence, find the maximum value $\displaystyle \alpha^2+\beta^2$.

2a. If $\displaystyle p(x)=3x^2+5x-2$ is a factor of the polynomial $\displaystyle q(x)=3x^4+8x^3+kx^2+3x-2$, determine the value of k. With this value of k, find all four zeroes of $\displaystyle q(x)$.

2b. Find the solution set of the inequality $\displaystyle x[p(x)]>0$.

Answer:

1. $\displaystyle \{k:-\frac{11}{3}\leq k \leq -1\}$; Maximum value $\displaystyle = \frac {200}{9}$

2a. $\displaystyle k=6; -2, \frac{1}{3}, -\frac{1}{2}\pm \frac{\sqrt{3}}{2}i$

2b. $\displaystyle \{x:-2<x<0$ or $\displaystyle x>\frac{1}{3}\}$