Proof of a conjecture on unimodality

Let P(x) be a polynomial of degree m, with nonnegative and nondecreasing coefficients. We settle the conjecture that for any positive real number d, the coefficients of P(x+d) form a unimodal sequence, of which the special case d being a positive integer has already been asserted in a previous work. Further, we explore the location of modes of P(x+d) and present some sufficient conditions on m and d for which P(x+d) has the unique mode ⌈m−dd+1⌉.