4 The polynomials \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are defined by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } + a x - 15 \quad \text { and } \quad \mathrm { q } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + b x + 21 ,$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\) and also of \(\mathrm { q } ( x )\).

1. Find the values of \(a\) and \(b\).
2. Show that the equation \(\mathrm { q } ( x ) - \mathrm { p } ( x ) = 0\) has only one real root.
   \includegraphics[max width=\textwidth, alt={}, center]{e2b16207-2cb7-412b-ba7f-758e4d3f1ffb-06_631_643_260_749} The diagram shows the curve \(y = 4 e ^ { - 2 x }\) and a straight line. The curve crosses the \(y\)-axis at the point \(P\). The straight line crosses the \(y\)-axis at the point \(( 0,9 )\) and its gradient is equal to the gradient of the curve at \(P\). The straight line meets the curve at two points, one of which is \(Q\) as shown.