Answer
$(x+y+5)(x^2+2xy+y^2-5x-5y+25)$
Work Step by Step
Using $a^3+b^3=(a+b)(a^2-2ab+b^2)$ or the factoring of the sum of 2 cubes, then,
\begin{array}{l}
(x+y)^3+125
\\=
[(x+y)+(5)][(x+y)^2-(x+y)(5)+(5)^2]
\\=
[(x+y)+(5)][(x^2+2xy+y^2)-5x-5y+25]
\\=
(x+y+5)(x^2+2xy+y^2-5x-5y+25)
.\end{array}
