(x+7)^3 = 49(x+7)
Expand the left side:
(x+7)(x+7)(x+7) = 49(x+7)
(x^2 + 14x + 49)(x+7) = 49(x+7)
x^3 + 7x^2 + 14x^2 + 98x + 49x + 343 = 49x + 343
x^3 + 21x^2 + 147x + 343 = 49x + 343
x^3 + 21x^2 + 147x = 49x
x^3 + 21x^2 + 98x = 0
x(x^2 + 21x + 98) = 0
x(x+7)(x+14) = 0

The solutions are x = 0, x = -7, x = -14.

x^3 + 5x^2 - x - 5 = 0
(x^3 + 5x^2) - (x + 5) = 0
x^2(x + 5) - 1(x + 5) = 0
(x^2 - 1)(x + 5) = 0
(x+1)(x-1)(x+5) = 0

The solutions are x = -5, x = -1, x = 1.

x^3 + 3x^2 = 16x + 48
x^3 + 3x^2 - 16x - 48 = 0

This is a cubic equation and may not have nice integer solutions. You can use numerical methods or a graphing calculator to approximate the solutions.