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The optimal exponential-quadratic control problem and exponential mean-square filtering problems are considered for stochastic Gaussian systems with polynomial first degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed form optimal control and filtering algorithms are obtained using quadratic value functions as solutions to the corresponding Hamilton-Jacobi-Bellman equations. The performance of the obtained risk-sensitive regulator and filter for stochastic first degree polynomial systems is verified in a numerical example against the conventional linear-quadratic regulator and Kalman-Bucy filter, through comparing the exponential-quadratic and exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithms in regard to the final criteria values.