Show that \(5x^2+nx-1=0\) has \(2\) real roots for all real values of \(n\).[2]
Let \(\mathrm{P}(x)=(x^2-6x+m^2)(5x^2+nx-1)\) for some real constants \(m\) and \(n\).
Find the range of values of \(m\) for which \(\mathrm{P}(x)=0\) yields only real roots.[4]
If the sum of the roots of \(\mathrm{P}(x)=0\) is \(10\) and one of the roots is \(3-\sqrt7\,\mathrm{i}\), where \(\mathrm{i}^2=-1\),
find the value of \(n\); and,
find the possible values of \(m\).[6]
Find the remainder when \(\mathrm{P}(x)\) is divided by \(2x-1\), leaving your answer in terms of \(m\) and \(n\).[2]
Solve the inequality \(\mathrm{P}(x)<0\) for \(m=n=3\).[4]