2023 SCSS P2 Q5

The polynomial \(\mathrm p(x)\) is given by \(\mathrm p(x)=ax^3+6x^2+bx-18\), where \(a\) and \(b\) are constants. It is given that \(x+2\) is a factor of \(\mathrm p(x)\) and when \(\mathrm p(x)\) is divided by \(x-3\) the remainder is 75.

1. Show that \(a=2\) and find the value of \(b\).[4]
2. Using the values from part (i), find the remainder when \(\mathrm p(x)\) is divided by \(2x-1\).[2]