you must choose the Hausman test result. the Hausman test is used to determine the appropriate model that should be applied to the panel data. Is it the random effects model or the fixed effects model?. while, the application of the Breusch–Pagan Lagrange multiplier test can be considered an additional step to detect the existence of random effects for the models indicated by Hausman test as having random effects in the regression analysis. by another words, Breusch–Pagan Lagrange multiplier test used to determine the type of the study model Is it the random effects model or the pooled OLS model?.
Thus, regarding to your results the Hausman test refers that the fixed effect is the most appropriate to be used thus no need to check the Breusch–Pagan Lagrange multiplier test.
All answers are correct, but u have to know first if the population of ur study is random or not. If its random, so First u have to apply Hausman test to chose between fixed and random effect model, if the p-value less than 5%, so u can chose Fixed effect model, but if its more than 5% u have to go further and apply Breusch–Pagan Lagrange multiplier test is to determine whether random or pooled model is more appropriate to ur study.
We use the hausman test to choose between fixed and random effects, whilst the Langrange multiplier test to choose between the OLS and the random effects. So it will depend on what actually you are looking for.