The problem asks us to simplify the expression $(\sqrt{8x-3}+3)^2$, assuming that all variables represent nonnegative numbers.

1. Problem Description

The problem asks us to simplify the expression $(\sqrt{8x-3}+3)^2$, assuming that all variables represent nonnegative numbers.

2. Solution Steps

We need to expand the square of the binomial. Recall the formula:

$(a+b)^2 = a^2 + 2ab + b^2$

In our case, $a = \sqrt{8x-3}$ and $b = 3$.

So, we have

$(\sqrt{8x-3}+3)^2 = (\sqrt{8x-3})^2 + 2(\sqrt{8x-3})(3) + 3^2$

$(\sqrt{8x-3})^2 = 8x-3$

$2(\sqrt{8x-3})(3) = 6\sqrt{8x-3}$

$3^2 = 9$

Therefore,

$(\sqrt{8x-3}+3)^2 = 8x-3 + 6\sqrt{8x-3} + 9$

Combining like terms, we get

$(\sqrt{8x-3}+3)^2 = 8x + 6 + 6\sqrt{8x-3}$

3. Final Answer

$8x+6+6\sqrt{8x-3}$