Can we factorise $6x^3+5x^2-17x-6$ completely?

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1. Given that \(f(x)=6x^3+5x^2-17x-6\),

   1. find the remainder when \(f(x)\) is divided by \(x-2\),
   2. find the remainder when \(f(x)\) is divided by \(x+2\),

Would it help us to recall the remainder theorem?

\(\qquad\) iii. factorise \(f(x)\) completely.

Can we find a remainder that is zero? What does this tell us about the roots of \(f(x)\)?

2. Given that the expression \(ax^3-x^2+bx+18\) is exactly divisible by \(x^2+x-6\), find the value of \(a\) and \(b\).

Would it help to try to factorise \(x^2+x-6\)?