To solve this equation, we first need to simplify it using logarithmic properties.

Given:
lg(127+x^3) - 3lg(x+1) = 0

Using the logarithmic properties, we can rewrite the equation as:
lg(127+x^3) - lg((x+1)^3) = 0

Now, we can combine the logarithms using the properties of logarithms:
lg((127+x^3)/(x+1)^3) = 0

To remove the logarithm, we can rewrite the equation as an exponential equation:
10^0 = (127+x^3)/(x+1)^3
1 = (127+x^3)/(x+1)^3

Now, we can simplify further:
(x+1)^3 = 127 + x^3
x^3 + 3x^2 + 3x + 1 = 127 + x^3
3x^2 + 3x - 126 = 0

Now, we have a quadratic equation that we can solve using factoring, the quadratic formula, or any other method.
Factoring out a 3, we get:
3(x^2 + x - 42) = 0
(x^2 + x - 42) = 0

Factoring the quadratic equation further, we get:
(x+7)(x-6) = 0

Setting each factor to 0 and solving for x, we get:
x+7 = 0
x = -7

and
x-6 = 0
x = 6

Therefore, the solutions to the equation lg(127+x^3) - 3lg(x+1) = 0 are x = -7 and x = 6.