The given equation is:

$9x^2 + 4x - 5 = 0$

This is a quadratic equation. To solve it, we can either factor the equation, use the quadratic formula, or complete the square. In this case, let's use the quadratic formula:

The quadratic formula is given by:

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

Here, $a = 9$, $b = 4$, and $c = -5$. Let's substitute these values into the formula:

$x = \frac{-4 \pm \sqrt{4^2 - 4(9)(-5)}}{2(9)}$

Now, solving the discriminant:

$b^2 - 4ac = 4^2 - 4(9)(-5) = 16 + 180 = 196$

Now we plug this back into the formula:

$x = \frac{-4 \pm \sqrt{196}}{18}$ $x = \frac{-4 \pm 14}{18}$

This gives two solutions:

1. $x_1 = \frac{-4 + 14}{18} = \frac{10}{18} = \frac{5}{9}$
2. $x_2 = \frac{-4 - 14}{18} = \frac{-18}{18} = -1$

Thus, the solutions are:

$x_1 = \frac{5}{9}, \quad x_2 = -1$

Would you like more details or have any questions?

Here are five related questions to expand on this:

1. Can we solve this quadratic equation by factoring instead of using the quadratic formula?
2. What happens if the discriminant (under the square root) is negative?
3. How do we graph the quadratic equation and find its vertex?
4. Can we apply the method of completing the square to this equation? How?
5. How do the solutions change if we modify the coefficients of the quadratic equation?

Tip: Always check the discriminant first—it helps determine whether the solutions will be real or complex!