<s>[INST] Let $c$ be a positive real number for which the equation $x^4-x^3+x^2-(c+1)x-(c^2+c)=0$ has a real root $\alpha$. Prove that $c=\alpha ^2 - \alpha$ Let $c$ be a positive real number for which the equation
$x^4-x^3+x^2-(c+1)x-(c^2+c)=0$
has a real root $\alpha$. Prove that $c=\alpha ^2 - \alpha$
I tried to to solve using relation between roots and coefficients but unable to progress much. Please help. Thanks in advance.
 [/INST] Direct substitution with $\alpha=x$ and $c=x^2-x$ leads to.
$x^4-x^3+x^2-x(x^2-x+1)-((x^2-x)^2+x^2-x)$
$=(1-1)x^4+(2-2)x^3+(2-2)x^2+(1-1)x=0$
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