- A curve has equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$
The curve touches the \(x\)-axis at the point \(P\) and crosses the \(x\)-axis at the point \(Q\).
- State the coordinates of the point \(P\).
- Find \(f ^ { \prime } ( x )\).
- Hence show that the equation of the tangent to the curve at the point where \(x = \frac { 5 } { 2 }\) can be expressed in the form \(y = k\), where \(k\) is a constant to be found.
The curve with equation \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, passes through the origin \(O\).
- State the possible values of \(a\).
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