Chapter 3.1, Problem 23E

Let a, b, c be real numbers and define $f:\text{R}\to \text{R}$ by $f\left(x\right)=x^3+a{x}^2+bx+c$. Prove that f is one-to-one if and only if $x_1\ne x_2$ implies $x_1^2+x_1{x}_2+x_2^2+a\left(x_1+x_2\right)+b\ne 0$.