Let's start by expanding the left side of the equation:

(5x-3)(x+2) - (x+4)²

= 5x(x) + 5x(2) - 3(x) - 3(2) - (x+4)(x+4)

= 5x² + 10x - 3x - 6 - (x² + 4x + 4x + 16)

= 5x² + 10x - 3x - 6 - x² - 8x - 16

= 4x² - x - 22

Now, we solve for x by setting the equation equal to zero:

4x² - x - 22 = 0

This is a quadratic equation, so we can use the quadratic formula to find the solutions for x:

x = (-(-1) ± sqrt((-1)² - 4(4)(-22))) / 2(4)
x = (1 ± sqrt(1 + 352)) / 8
x = (1 ± sqrt(353)) / 8

Therefore, the solutions for x are:

x = (1 + sqrt(353)) / 8 and x = (1 - sqrt(353)) / 8