Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local holomorphic function $f$ and local section $s$. Locally such conections look like $$ D = \partial + A, $$ where $A$ is a holomorphic section of $\operatorname{End}(E)\otimes \Omega_X$.
If $D$ is a holomorphic connection on $E$ then $D+\bar{\partial}$ is an "ordinary" connection on $E$.
How can one show that if $E$ admits a holomorphic connection then it admits a flat connection? In other words there is $B \in\operatorname{End}(E)\otimes \Omega_X$ s.t. $$ (\bar{\partial} +D+B)^2=0. $$