Let's solve the equation $3x^2 = 5 - 2x$ and find the even numbers among the solutions.

1. Rewrite the equation in standard form:
   $3x^2 + 2x - 5 = 0$

2. Solve the quadratic equation using the quadratic formula:
   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 3$, $b = 2$, and $c = -5$.

3. Calculate the discriminant:
   $b^2 - 4ac = 2^2 - 4(3)(-5) = 4 + 60 = 64$

4. Find the roots:
   $x = \frac{-2 \pm \sqrt{64}}{2(3)}$
   $x = \frac{-2 \pm 8}{6}$

5. Calculate the two solutions:
   $x_1 = \frac{-2 + 8}{6} = \frac{6}{6} = 1$
   $x_2 = \frac{-2 - 8}{6} = \frac{-10}{6} = -\frac{5}{3}$

The solutions are $x = 1$ and $x = -\frac{5}{3}$. Among these, the only integer value is $x = 1$. Since 1 is odd, there are no even integer solutions to the given equation.

Would you like a more detailed explanation of the steps or have any questions?

Here are some related questions for further exploration:

1. How do you determine if a quadratic equation has real or complex solutions?
2. What are the properties of quadratic equations with a discriminant of zero?
3. Can a quadratic equation have exactly one real root?
4. What is the sum of the roots of the quadratic equation $3x^2 + 2x - 5 = 0$?
5. How do you factorize a quadratic expression if the roots are not integers?
6. What are some practical applications of solving quadratic equations?
7. How does the graph of a quadratic function relate to its equation's solutions?
8. What is the axis of symmetry for the parabola represented by $3x^2 + 2x - 5 = 0$?

Tip: When using the quadratic formula, always check the discriminant first to determine the nature of the roots (real or complex).