To begin solving this equation, we can first simplify the equation by using exponential rules.

We know that 7 = 3 + 4, so we can rewrite 7^(x+1) as (3 + 4)^(x+1). Expanding this using the binomial theorem, we get:

(3 + 4)^(x+1) = 3^(x+1) + (x+1)3^x4 + ((x+1)(x+1-1))/2 3^(x-1) * 4^2 + ...

Let's simplify further:

3^(x+3) + 3^x = 3^(x+1) + (x+1)3^x4 + ((x+1)(x))/2 3^(x-1) * 4^2 + ...

Now, we can substitute this back into the original equation:

3^(x+3) + 3^x = 3^(x+1) + (x+1)3^x4 + ((x+1)(x))/2 3^(x-1) 4^2 + ... + 57^x

Now, we have a messy equation that includes higher powers of 3 and 4. We can continue simplifying it, but it may not lead to an elegant solution.

Alternatively, we can use numerical methods to solve this equation by graphing it and finding the approximate values of x where the equation is true.