Q.Find the transformed equation whose roots are the negatives of the roots of $x^4+5x^3+11x+3=0$.

Replace $x$ by $-x$ in the given equation; if $\alpha$ solves the original, $-\alpha$ solves the transformed one.

Let $f(x)=x^4+5x^3+11x+3$ (no $x^2$ term, i.e. coefficient of $x^2$ is $0$). If $\alpha$ is a root of $f(x)=0$, we want $g(x)=0$ whose roots are $-\alpha$. Take $g(x)=f(-x)$: then $g(-\alpha)=f(\alpha)=0$, exactly as required.

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