To solve this equation, we can start by isolating the square root term on one side of the equation.

3x - 5/√x + 4 = √x + 4

First, let's multiply both sides of the equation by √x + 4 to eliminate the denominator:

(3x - 5/√x + 4)(√x + 4) = (√x + 4)(√x + 4)

Expanding both sides:

3x(√x + 4) - 5 = (√x + 4)(√x + 4)

Now, distribute on the left side:

3x√x + 12x - 5 = (√x + 4)(√x + 4)

Next, distribute on the right side:

3x√x + 12x - 5 = x + 4√x + 4√x + 16

Combine like terms:

3x√x + 12x - 5 = x + 8√x + 16

Now, let's move all the terms involving √x to one side of the equation:

3x√x - x - 8√x = 16 - 12x + 5

Combine like terms:

(3x - 1)√x - 8√x = 21 - 12x

Now, factor out the common factor of √x:

(3x - 1 - 8)√x = 21 - 12x

Simplify:

-5√x = 21 - 12x

Next, let's isolate the square root term by moving the -12x term to the other side:

-5√x + 12x = 21

Now, let's square both sides of the equation to eliminate the square root:

(-5√x + 12x)^2 = 21^2

Expanding both sides:

25x - 60√x + 144x^2 = 441

Now, let's move all the terms to one side of the equation:

144x^2 + 25x - 60√x - 441 = 0

This is a quadratic equation in terms of x. To solve it, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 144, b = 25, and c = -441. Plugging these values into the quadratic formula:

x = (-25 ± √(25^2 - 4(144)(-441))) / (2(144))

Simplifying:

x = (-25 ± √(625 + 253440)) / 288

x = (-25 ± √253065) / 288

Since the square root of 253065 is not a perfect square, we cannot simplify it further. Therefore, the solutions for x are:

x = (-25 + √253065) / 288

x = (-25 - √253065) / 288