To find the zeros of the quadratic function $f(x) = 2x^2 + 16.8x + 31$, we need to solve for $x$ where $f(x) = 0$. The equation to solve is:

$2x^2 + 16.8x + 31 = 0$

We can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 2$, $b = 16.8$, and $c = 31$.

Step 1: Calculate the discriminant

The discriminant is given by $b^2 - 4ac$:

$b^2 - 4ac = (16.8)^2 - 4(2)(31) = 282.24 - 248 = 34.24$

Step 2: Apply the quadratic formula

Since the discriminant is positive, we have two real solutions:

$x = \frac{-16.8 \pm \sqrt{34.24}}{2(2)}$

Calculating $\sqrt{34.24} \approx 5.85$, the solutions for $x$ are:

$x = \frac{-16.8 + 5.85}{4} \approx \frac{-10.95}{4} \approx -2.7375$

$x = \frac{-16.8 - 5.85}{4} \approx \frac{-22.65}{4} \approx -5.6625$

Rounding to the nearest thousandth, the zeros are approximately:

$x \approx -2.738 \quad \text{and} \quad x \approx -5.663$

Would you like more details on this calculation or have any questions?

Here are five related questions to expand your understanding:

1. How can the quadratic formula be derived from completing the square?
2. What does the discriminant tell us about the nature of the roots?
3. How would you find the zeros if the discriminant were negative?
4. Can the quadratic formula be applied to solve higher-degree polynomials?
5. How do transformations of quadratic functions affect their zeros?

Tip: Always check the discriminant first to determine the number and type of roots you can expect (real or complex).