Squaring both sides of sqrt(13-x^2) = x+5 gives 13-x^2 = x^2+10x+25, which simplifies to 2x^2+10x+12 = 0 or x^2+5x+6 = 0. The roots are x = -2 and x = -3. Checking these in the original equation, sqrt(13-4) = 3 and -2+5 = 3 (valid), while sqrt(13-9) = 2 and -3+5 = 2 (valid). The product is (-2)*(-3) = 6.