7 A curve has parametric equations
$$x = \frac { 2 t + 3 } { t + 2 } , \quad y = t ^ { 2 } + a t + 1$$
where \(a\) is a constant. It is given that, at the point \(P\) on the curve, the gradient is 1 .
- Show that the value of \(t\) at \(P\) satisfies the equation
$$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1 = 0$$
- It is given that \(( t + 1 )\) is a factor of
$$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1$$
Find the value of \(a\).
- Hence show that \(P\) is the only point on the curve at which the gradient is 1 .
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