Think outside the box

Calculus Level 4

Let \(f^{(0)}(x) = (1+2x-x^2)^4\), where \(f^{(n)}(x)\) denotes the \(n^\text{th}\) derivative of \(f(x)\). If the value of \(f^{(7)}(0)\) is in the form \(-k!\), where \(k\) is a natural number, find the value of \(k\).

\[\]Notation: \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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