Find Possible values of $c$ such that polynomial $$x(x+1)(x+2) \cdots (x+2009)=c$$ has one root of multiplicity two
My Try:
Assuming the polynomial equation has root $k$ Repeated twice The given polynomial can be written as
$$f(x)=x(x+1)(x+2)\cdots (x+2009)-c =(x+k)^2 \left(x^{2008}+a_1x^{2007}+a_2x^{2006}+\cdots-\frac{c}{k^2}\right)$$
Now $f'(x)=0$ should have $k$ as a root with multiplicity One
we have
$$f'(x)=f(x)\left(\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2}+\cdots+\frac{1}{x+2009}\right)$$
can we use this concept to find $c$