Simplify (x/3-4)² + x(√8x/3+2)(√8x/3-2): Complete Solution

Let's solve this problem by simplifying each component separately:

First, simplify $(\frac{x}{3} - 4)^2$ using the square of a difference formula:

$(\frac{x}{3} - 4)^2 = \left(\frac{x}{3}\right)^2 - 2 \times \frac{x}{3} \times 4 + 4^2$ This becomes:

$= \frac{x^2}{9} - \frac{8x}{3} + 16$

Next, simplify $x(\frac{\sqrt{8x}}{3} + 2)(\frac{\sqrt{8x}}{3} - 2)$ using the difference of squares formula:

$(\frac{\sqrt{8x}}{3} + 2)(\frac{\sqrt{8x}}{3} - 2) = \left(\frac{\sqrt{8x}}{3}\right)^2 - 2^2$ Simplify further:

$= \frac{8x}{9} - 4$

Including the factor of $x$, we have:

$x \left(\frac{8x}{9} - 4\right) = \frac{8x^2}{9} - 4x$

Combine the results from both parts:

$\left(\frac{x^2}{9} - \frac{8x}{3} + 16\right) + \left(\frac{8x^2}{9} - 4x\right)$

Simplify by combining like terms:

$= \frac{x^2}{9} + \frac{8x^2}{9} - \frac{8x}{3} - 4x + 16 = x^2 - \left(\frac{8x}{3} + 4x\right) + 16 = x^2 - \left(\frac{8x + 12x}{3}\right) + 16 = x^2 - \frac{20x}{3} + 16$

Therefore, after simplifying, the expression becomes $\boldsymbol{x^2 - \frac{20x}{3} + 16}$.

The final solution is: $x^2 - \frac{20x}{3} + 16$.