The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicativenon-Gaussian noise and additive Gaussian white noise is investigated.Firstly,the non-Markov process is reduced tothe Markov process through a path-integral approach;Secondly,the approximate Fokker-Planck equation is obtainedby applying the unified coloured noise approximation,the small time delay approximation and the Novikov Theorem.The functional analysis and simplification are employed to obtain the approximate expressions of MFPT.The effects ofnon-Gaussian parameter (measures deviation from Gaussian character) r,the delay time τ,the noise correlation timeτ_0,the intensities D and α of noise on the MFPT are discussed.It is found that the escape time could be reduced byincreasing the delay time τ,the noise correlation time τ_0,or by reducing the intensities D and α.As far as we know,thisis the first time to consider the effect of delay time on the mean first-passage time in the stochastic dynamical system. The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicative non-Gaussian noise and additive Gaussian white noise is investigated. Firstly, the non-Markov process is reduced tothe Markov process through a path- the approximate Fokker-Planck equation is obtained by applying the unified colored noise approximation, the small time delay approximation and the Novikov Theorem. The functional analysis and simplification are employed to obtain the approximate expressions of MFPT. The effects ofnon-Gaussian parameter (measures deviation from Gaussian character) r, the delay time τ, the noise correlation timeτ_0, the intensities D and α of noise on the MFPT are discussed. It is found that the escape time could be reduced by incrementing the delay time τ, the noise correlation time τ_0, or by reducing the intensities D and α. As far as we know, thisis the first time to consider the effect of delay time on the mean first-passage time in the s tochastic dynamical system.