Polynomials, Cubic Equations and Partial Fractions - Factorisation by Special Identities

It is given that \(\mathrm{f}\left( x \right)={{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}-2x+1\) and \(\mathrm{g}\left( x \right)=3{{x}^{4}}-48\), where \(a\) and \(b\) are integers.

1. If \(\mathrm{f}\left( x \right)\) has a remainder of \(2x-3\) when it is divided by \({{x}^{2}}-4\), find the value of \(a\) and of \(b\).
2. Factorise \(\mathrm{g}\left( x \right)\) completely. Hence state the real values of \(x\) if \(\mathrm{g}\left( x \right)=0\).
3. Given further that \(2\mathrm{f}\left( x \right)-3\mathrm{g}\left( x \right)=\mathrm{Q}\left( x \right)\left( x+2 \right)+\mathrm{R}\left( x \right)\), where \(\mathrm{Q}\left( x \right)\) and \(\mathrm{R}\left( x \right)\) are the quotient and remainder respectively, find \(\mathrm{R}\left( x \right)\).