To solve this logarithmic equation, we need to use the properties of logarithms to condense the terms and then isolate the variable.

Given equation:

log2^2 (x+4) + 4log2 (2x) - 9 = 0

First, apply the power property of logarithms, loga^n = nloga:

2(log2 (x+4)) + log2 (16x) - 9 = 0

Next, we can condense the terms by combining the logarithms with the same base:

log2 ((x+4)^2) + log2 (16x^4) - 9 = 0

Now, we can combine the logarithms into a single logarithm using the product property of logarithms, loga (x) + loga (y) = loga (xy):

log2((x+4)^2 * 16x^4) - 9 = 0

log2(16(x+4)^2*x^4) - 9 = 0

Now, convert the equation to exponential form:

2^(log2(16(x+4)^2*x^4)) = 2^9

16(x+4)^2*x^4 = 512

16(x^2 + 8x + 16)*x^4 = 512

16x^6 + 128x^5 + 256x^4 = 512

Divide the equation by 16:

x^6 + 8x^5 + 16x^4 = 32

Now, we need to solve this polynomial equation for x. This can be a bit complicated and may require numerical methods or factoring techniques.