To solve this logarithmic equation, we can use the properties of logarithms to combine the terms on the left side of the equation and simplify it.

First, we need to combine the two logarithmic terms using the product rule of logarithms, which states that log(a) + log(b) = log(ab):

lg (x+2) + lg (10x+20) = lg ((x+2)(10x+20))

Next, we can simplify the expression inside the logarithm by multiplying the terms:

lg ((x+2)(10x+20)) = lg (10x^2 + 40x + 20)

Now, we have the equation in the form of lg (10x^2 + 40x + 20) = 3. To solve for x, we can convert the logarithmic equation into an exponential equation:

10x^2 + 40x + 20 = 10^3

Simplify the right side of the equation:

10x^2 + 40x + 20 = 1000

Now we have a quadratic equation that we can solve using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Plug in the values of a, b, and c into the formula:

x = (-40 ± √(40^2 - 41020)) / 2*10

x = (-40 ± √(1600 - 800)) / 20

x = (-40 ± √(800)) / 20

x = (-40 ± 28.28) / 20

This gives us two possible solutions for x:

x = (-40 + 28.28) / 20 ≈ -0.584
x = (-40 - 28.28) / 20 ≈ -3.142

So, the solutions to the equation lg (x+2) + lg (10x+20) = 3 are x ≈ -0.584 and x ≈ -3.142.