I want to show
Px$(B(s)\ge0 $ for all 0 $\le s \le t$ and B(t) $\in$ M) = Px(B(t) \in M)$-$P-x$(B(t) \in M)$
where,x>0,M is measurable set in [0,$\infty$).
The difficulty for me is how to handle the left side of the equation.I know the distribution of hitting time.But here it is not only about hitting time.
This is called the reflection principle (often named after Désiré André) and is most probably explained in your notes: one starts from the decomposition $$ [B_t\in M]=[B_t\in M,T_0\lt t]\cup[B_t\in M,T_0\gt t]. $$ where $T_0=\inf\{s\gt0\mid B_s=0\}$ is the first hitting time of $0$. Once one has identified the first and the third event above, one is left with the crucial remark that, under $P_x$, the process $B'$ defined by $$ B_s'=\left\{\begin{array}{cll}B_s&\text{if}&s\leqslant T_0\\-B_s&\text{if}&s\gt T_0\end{array}\right. $$ is distributed like $(B_s)$ and that $\inf\{s\gt0\mid B_s'=0\}=T_0$. Furthermore, for every $x\gt0$, $M\subseteq[0,\infty)$ hence the events $[B_0=x,B_t\in M,T_0\lt t]$ and $[B_0'=x,B_t'\in-M]$ coincide. It follows that $$ P_x(B_t\in M,T_0\lt t)=P_x(B_t'\in-M)=P_x(B_t\in-M), $$ which ends the proof of the formula in your question.