$4 x^{2}+16 x-9=0$

Step 1 ：Given the quadratic equation \(4 x^{2}+16 x-9=0\)

Step 2 ：This is a quadratic equation of the form \(ax^2 + bx + c = 0\)

Step 3 ：The solutions to this equation can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Step 4 ：Let a = 4, b = 16, c = -9

Step 5 ：Calculate the discriminant D = \(b^2 - 4ac = 400\)

Step 6 ：Find the roots of the equation using the quadratic formula: \(x1 = -4.5\), \(x2 = 0.5\)

Step 7 ：Final Answer: The solutions to the equation \(4 x^{2}+16 x-9=0\) are \(\boxed{-4.5}\) and \(\boxed{0.5}\)