Expanding the left side of the equation:

(4x + 5)² = (4x + 5)(4x + 5)
= 16x² + 20x + 20x + 25
= 16x² + 40x + 25

Substitute this back into the equation:

16x² + 40x + 25 - 13x² = 100 + 40x
3x² + 40x + 25 = 100 + 40x

Rearranging terms:

3x² - 15x - 75 = 0

Factor out 3:

3(x² - 5x - 25) = 0

Now, solving for x using the quadratic formula:

x = [5 ± sqrt(5² - 4(1)(-25))] / 2
x = [5 ± sqrt(25 + 100)] / 2
x = [5 ± sqrt(125)] / 2
x = [5 ± 5√5] / 2

Therefore, the solutions for x are:

x₁ = 5/2 + 5/2√5
x₂ = 5/2 - 5/2√5