Finding the Zeros of the Polynomial
To find the zeros of the polynomial \(4x^2 + 5\sqrt{2}x - 3\), we need to set the polynomial equal to zero and solve for x. The zeros of a polynomial are the values of x that make the polynomial equal to zero.

Setting up the Equation
We have the polynomial \(4x^2 + 5\sqrt{2}x - 3\). We set this equal to zero to find the zeros:
\(4x^2 + 5\sqrt{2}x - 3 = 0\)

Using the Quadratic Formula
To solve for x, we can use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where a, b, and c are the coefficients of the polynomial.

Identifying Coefficients
In our polynomial, a = 4, b = 5√2, and c = -3. Plugging these values into the quadratic formula, we get:
\(x = \frac{-5\sqrt{2} \pm \sqrt{(5\sqrt{2})^2 - 4(4)(-3)}}{2(4)}\)

Solving for Zeros
Calculating the discriminant and simplifying further will give us the zeros of the polynomial:
\(x = \frac{-5\sqrt{2} \pm \sqrt{50 + 48}}{8}\)
\(x = \frac{-5\sqrt{2} \pm \sqrt{98}}{8}\)
Therefore, the zeros of the polynomial are:
\(x = \frac{-5\sqrt{2} + \sqrt{98}}{8}\) and \(x = \frac{-5\sqrt{2} - \sqrt{98}}{8}\)