Solve the Quadratic Inequality: -x² - 9 > 0

To solve this quadratic inequality, $-x^2 - 9 > 0$, we will follow these steps:

• Step 1: Identify the quadratic expression $-x^2 - 9$.
• Step 2: Attempt transformation and determine when the expression $-x^2 - 9$, can be greater than zero.

Let's analyze the equation:

Rewrite the inequality:
$-x^2 - 9 > 0$

Add 9 to both sides:
$-x^2 > 9$

Multiply the entire inequality by $-1$ and remember to reverse the inequality sign:
$x^2 < -9$

Observe the inequality $x^2 < -9$:
Note that $x^2$, being a square of any real number, is always greater than or equal to zero.

As $x^2$ cannot be less than negative nine for any real number $x$, the inequality has no solution in the realm of real numbers.

Therefore, the correct answer is:

There is no solution.