It is given that

\(p(x)=2x^3+ax^2+4x+b,\)

where \(a\) and \(b\) are constants. It is given also that \(2x+1\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \(x-1\) there is a remainder of \(-12\).

(i) Find the value of \(a\) and of \(b\).

(ii) Using your values of \(a\) and \(b\), write \(p(x)\) in the form \((2x+1)q(x)\), where \(q(x)\) is a quadratic expression.

(iii) Hence find the exact solutions of the equation \(p(x)=0\).