Given that $_{ }$$x^{4}+x^{3}+px^{2}+4x-2=0$ where $p$ is a constant, has roots $x_{1}, x_{2}, x_{3}\,and\,x_{4}$
a) Find the equation whose roots are $\frac{1}{x_{1}}, \frac{1}{x_{2}}, \frac{1}{x_{3}}\,and\,\frac{1}{x_{4}}$
b) Given that $ x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{3}^{2}=\frac{1}{x_{1}^{2}}+\frac{1}{x_{2}^{2}}+\frac{1}{x_{3}^{2}}+\frac{1}{x_{4}^{2}} $, find the value of $p$.
I know (from Vieta's formulas) that the sum of roots = -1 and product of roots = -2. I feel that it can be solved using Vieta's theorem but I am stuck. Please help with a hint on how to proceed. Thank you for any help you can offer.
Hints: (a) if $P(x)$ is a polynomial of degree $n$ in $x$ with roots $x_k \ne 0$, then $Q(x)=x^nP\left(\frac{1}{x}\right)$ is a polynomial of degree $n$ with roots $\frac{1}{x_k}$. (b) $x_1^2+x_2^2+x_3^2+x_4^2=\left(\sum_k x_k\right)^2 - 2\,\sum_{i \lt j} x_ix_j\,$.