To solve this equation, we need to first simplify it by expanding the expression inside the logarithm:

4^x - 2^x + 1 + 2 = x

Now, we can rewrite the equation in terms of log base 2:

log2(4^x) - log2(2^x) + log2(1) + log2(2) = x

Using the properties of logarithms, we can simplify this equation further:

xlog2(4) - xlog2(2) + log2(1) + 1 = x

Now, we can substitute the values of log2(4) and log2(2) into the equation:

x2 - x1 + 0 + 1 = x

2x - x + 1 = x
x + 1 = x

Since this equation does not make sense (the variable x cannot be equal to x+1), it is possible that there was a mistake in the simplification process. Let me know if you still need assistance with this equation.