To build $ P(x), $ we start with the equation $ x = \sqrt{2} + \sqrt{3} + \sqrt{5} $ and repeatedly rearrange and square the equation until all the terms have rational coefficients. First, we subtract $ \sqrt{5} $ from both sides, giving \[x - \sqrt{5} = \sqrt{2} + \sqrt{3}.\]Then, squaring both sides, we have \[\begin{aligned} (x-\sqrt5)^2 &= 5 + 2\sqrt{6} \\ x^2 - 2x\sqrt{5} + 5 &= 5 + 2\sqrt{6} \\ x^2 - 2x\sqrt{5} &= 2\sqrt{6}.\end{aligned}\]Adding $ 2x\sqrt{5} $ to both sides and squaring again, we get \[\begin{aligned} x^2 &= 2x\sqrt{5} + 2\sqrt{6} \\ x^4 &= (2x\sqrt{5} + 2\sqrt{6})^2 \\ x^4 &= 20x^2 + 8x\sqrt{30} + 24.\end{aligned}\]To eliminate the last square root, we isolate it and square once more: \[\begin{aligned} x^4 - 20x^2 - 24 &= 8x\sqrt{30} \\ (x^4 - 20x^2-24)^2 &= 1920x^2.\end{aligned}\]Rewriting this equation as \[(x^4-20x^2-24)^2 - 1920x^2 = 0,\]we see that $ P(x) = (x^4-20x^2-24)^2 - 1920x^2 $ is the desired polynomial. Thus, \[\begin{aligned} P(1) &= (1-20-24)^2 - 1920 \\ &= 43^2 - 1920 \\ &= \boxed{-71}.\end{aligned}\]