To solve this quadratic equation, we can use the quadratic formula:

The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a

In the given equation, a = 1, b = 16, and c = -3.

Plugging these values into the quadratic formula, we get:
x = (-16 ± √(16^2 - 41(-3))) / 2*1
x = (-16 ± √(256 + 12)) / 2
x = (-16 ± √268) / 2

Now we can simplify this expression further:
x = (-16 ± √268) / 2
x = (-16 ± √(4*67)) / 2
x = (-16 ± 2√67) / 2
x = -8 ± √67

Therefore, the solutions to the equation x^2 + 16x - 3 = 0 are:
x = -8 + √67 and x = -8 - √67