2025 IGCSE Additional Mathematics October/November 0606/12 Q2

The polynomial \(\mathrm p\) is such that \(\mathrm p(x)=2x^3+ax^2+13x+b\), where \(a\) and \(b\) are integers.

It is given that \(x+2\) is a factor of \(\mathrm p(x)\).

When \(\mathrm p(x)\) is divided by \(x+1\) there is a remainder of 6.

1. Find the values of \(a\) and \(b\).[4]
2. Show that the equation \(\mathrm p(x)=0\) has only one real root.[3]