Let be a random algebraic polynomial where the coefficients A₀,A₁,… form a sequence of centered Gaussian random variables. Moreover, assume that the increments Δ_j=A_j-A_j-1,j=0,1,2,… are independent, A_-1=0. The coefficients A₀, A₁, … A_n can be considered as n consecutive observations of a Brownian motion. We provide an explicit formula for the expected density of the number of real zeros of Q_n(x). We observe that the expected density of real zeros on (-1/p,1/p) has the limit p/(1-p²x²), n→∞ where p^2j=Var(Δ_j), and we obtain the asymptotic behaviour of the expected number of real zeros for the case that p=1, which exhibits new features in the study of random algebraic polynomials.