There are many known asymptotic estimates of the expected number of zeros of a polynomial of degree $n$ with independent random coefficients, for $n \rightarrow \infty$. The present paper provides an estimate of the expected number of times that such a polynomial assumes the real value $K$, where $K$ is not necessarily zero. The coefficients are assumed to be normally distributed. It is shown that the results are valid even for $K \rightarrow \infty$, as long as $K = O(\sqrt n)$.
Kambiz Farahmand. "On the Average Number of Real Roots of a Random Algebraic Equation." Ann. Probab. 14 (2) 702 - 709, April, 1986. https://doi.org/10.1214/aop/1176992539