To solve the equation lg$x+1$ + lg$x+4$ = 1, we can combine the logarithms using the properties of logarithms.

lg$x+1$ + lg$x+4$ = lg$(x+1)(x+4)$

Now, we can rewrite the equation as:

lg$(x+1)(x+4)$ = 1

Using the definition of logarithms, we know that 10^1 = $x+1$$x+4$.

Therefore, we have:
10 = x^2 + 5x + 4

Rearranging the equation:
x^2 + 5x - 6 = 0

Now, we can factor the quadratic equation:
$x+6$$x-1$ = 0

Setting each factor to zero:
x + 6 = 0 or x - 1 = 0

x = -6 or x = 1

Therefore, the solutions to the equation lg$x+1$ + lg$x+4$ = 1 are x = -6 and x = 1.