#### Answer

$log_{10}(x^{2}+8x+15)$

#### Work Step by Step

We know that $log_{b}xy=log_{b}x+log_{b}y$ (where $x$, $y$, and $b$ are positive real numbers and $b\ne1$).
Therefore, $log_{10}(x+3)+log_{10}(x+5)=log_{10}((x+3)(x+5))$.
We can use the FOIL method to further simplify.
$log_{10}((x+3)(x+5))=log_{10}(x^{2}+5x+3x+15)=log_{10}(x^{2}+8x+15)$.