The expression \[2x^4+ax^3+bx^2-4x-4\] where \(a\) and \(b\) are constants, is denoted by \(f(x)\). Given that \(f\left(-\dfrac{1}{2}\right)=0\) and \(f(2)=0\), find the values of \(a\) and \(b\).

Could we find two equations involving \(a\) and \(b\) that must be simultaneously true?

With these values for \(a\) and \(b\)

- express \(f(x)\) as the product of three algebraic factors, and hence show that the equation \(f(x)=0\) has only two real roots;

How would the Factor Theorem help us here?

How can we find the number of real roots of a quadratic equation?

- find the set of values of \(x\) for which \(f(x)>0\).

Could sketching the graph of \(f(x)\) help with this?