To solve the quadratic equation 4x² + 4√3x + 1 = 0, we can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

In this case, a = 4, b = 4√3, and c = 1. Plugging these values into the formula:

x = [-4√3 ± √((4√3)² - 441)] / 2*4
x = [-4√3 ± √(48 - 16)] / 8
x = [-4√3 ± √32] / 8

Now, simplify the expression under the square root:
x = [-4√3 ± √(16*2)] / 8
x = [-4√3 ± 4√2] / 8

Now, combine like terms:
x = (-4√3 + 4√2) / 8 = (-4√3 + 4√2) / 8
x = -√3 + √2 / 2

So, the solutions to the quadratic equation 4x² + 4√3x + 1 = 0 are:

x = (-√3 + √2) / 2
x = (-√3 - √2) / 2