Given the inequality:
$$\frac{x^{2} \log{\left(- x + 4 \right)}}{\log{\left(512 \right)}} \geq \frac{\log{\left(x^{2} - 8 x + 16 \right)}}{\log{\left(2 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x^{2} \log{\left(- x + 4 \right)}}{\log{\left(512 \right)}} = \frac{\log{\left(x^{2} - 8 x + 16 \right)}}{\log{\left(2 \right)}}$$
Solve:
$$x_{1} = 3$$
$$x_{2} = -4.24264068711928$$
$$x_{1} = 3$$
$$x_{2} = -4.24264068711928$$
This roots
$$x_{2} = -4.24264068711928$$
$$x_{1} = 3$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-4.24264068711928 - \frac{1}{10}$$
=
$$-4.34264068711928$$
substitute to the expression
$$\frac{x^{2} \log{\left(- x + 4 \right)}}{\log{\left(512 \right)}} \geq \frac{\log{\left(x^{2} - 8 x + 16 \right)}}{\log{\left(2 \right)}}$$
$$\frac{\left(-4.34264068711928\right)^{2} \log{\left(4 - -4.34264068711928 \right)}}{\log{\left(512 \right)}} \geq \frac{\log{\left(16 + \left(-4.34264068711928\right)^{2} - 8 \left(-4.34264068711928\right) \right)}}{\log{\left(2 \right)}}$$
40.0061005618117 4.24275959081044
---------------- >= ----------------
log(512) log(2)
one of the solutions of our inequality is:
$$x \leq -4.24264068711928$$
_____ _____
\ /
-------•-------•-------
x_2 x_1
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -4.24264068711928$$
$$x \geq 3$$