Find the quotient and remainder when
$$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$
is divided by ( \(x ^ { 2 } - x + 4\) ).
It is given that, when
$$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$
is divided by ( \(x ^ { 2 } - x + 4\) ), the remainder is zero. Find the values of the constants \(p\) and \(q\).
When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation
$$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$
and state what that value is.