ISL 2017 A1

Let $a_{1}, a_{2}, \ldots, a_{n}, k$, and $M$ be positive integers such that $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \text { and } a_{1} a_{2} \ldots a_{n}=M$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \cdots\left(x+a_{n}\right)$$ has no positive roots. (Trinidad and Tobago)