a) We graph \(f(x) = x^{3} - 4x^{2} + 2x + 4\) using a graphing calculator to find the rational zeros.

It cuts the graph at \(x = 2\)

Means \(x = 2\) is a zero.

Then we use \(x=2\) to find the other factor. We use synthetic division.

Another factor is \(x^{2}-2x-2.\)

Then we use the quadratic formula to find the zeros from it

\(x^{2} - 2x - 2 = 0\)

\(x=\frac{2 \pm\sqrt{(-2)^{2}-4(1)(-2)}}{2(1)}\)

\(x=\frac{2 \pm \sqrt{12}}{2}\)

\(x=\frac{2 \pm 2 \sqrt{3}}{2}\)

\(x=1 \pm \sqrt{3}\)

Answer \((a): 1 -\sqrt{3},2,1+\sqrt{3}\)

b) Linear factored form of f is:

Answer: \((x-1+\sqrt{3})(x-2)(x-1-\sqrt{3})\)