In the following [K] refers to the paper http://dml.cz/bitstream/handle/10338.dmlcz/120545/ActaOstrav_07-1999-1_3.pdf.  

That a split exact complex of projectives $(P,d)$ is a projective object can be seen as follows: 

 1. $im(d_n)$ is projective since it is a direct summand of $P_{n-1}$ 

 2. By http://mathoverflow.net/questions/103056/when-is-an-acylic-chain-complex-contractible/103071#103071 a split exact complex is contractible, so $P$ is contractible. 

 3. By [K], Lemma 4.4 a contractible complex (like $P$) is isomorphic to the mapping cone of the boundary subcomplex 
$$ \cdots \to im(d_{n+1}) \xrightarrow{0} im(d_n) \to \cdots$$

 4. By [K], Theorem 3.1, the mapping cone of a complex of projectives with zero differentials is a projective object. Hence $P$ is a projective object by 1. and 3.