Answer: \(\mathrm{f}(x)=-\frac12(x+5)(x+1)(x-2)\); \(-5\lt x\lt -1\) or \(x\gt 2\).

(a) From the graph, the roots are

\(x=-5,\quad x=-1,\quad x=2.\)

Therefore

\(\mathrm{f}(x)=k(x+5)(x+1)(x-2).\)

The graph cuts the \(y\)-axis at \(5\), so \(\mathrm{f}(0)=5\). Hence

\(k(5)(1)(-2)=5.\)

So

\(-10k=5,\qquad k=-\frac12.\)

Thus

\(\mathrm{f}(x)=-\frac12(x+5)(x+1)(x-2).\)

(b) The graph is below the \(x\)-axis where \(\mathrm{f}(x)\lt 0\). From the sketch this happens between the first two roots and to the right of the last root:

\(-5\lt x\lt -1\quad\text{or}\quad x\gt 2.\)