The diagram shows the curve $y = {x^2}{e^{ - x}}$.
a) Show that the area of the shaded region bounded by the curve, the x-axis and the line $x = 3$ is equal to $2 - \frac{{17}}{{{e^3}}}$.
b) Find the x-coordinate of the maximum point $M$ on the curve.
c) Find the x-coordinate of the point $P$ at which the tangent to the curve passes through the origin.

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