To solve the quadratic equation \(-15x^2 + 216x - 84 = 0\), we can start by simplifying the equation. We can factor out \(-3\) from all terms:

\[
-3(5x^2 - 72x + 28) = 0
\]

Now, we can focus on solving the quadratic equation \(5x^2 - 72x + 28 = 0\). We can use the quadratic formula, which is given by:

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

In our case, \(a = 5\), \(b = -72\), and \(c = 28\). Now we calculate the discriminant (\(b^2 - 4ac\)):

\[
b^2 - 4ac = (-72)^2 - 4 \cdot 5 \cdot 28
\]

Calculating these values:

\[
(-72)^2 = 5184,
\]
\[
4 \cdot 5 \cdot 28 = 560,
\]
\[
b^2 - 4ac = 5184 - 560 = 4624.
\]

Now we can find the square root of 4624:

\[
\sqrt{4624} = 68.
\]

Substituting back into the quadratic formula:

\[
x = \frac{-(-72) \pm 68}{2 \cdot 5} = \frac{72 \pm 68}{10}.
\]

This gives us two possible solutions for \(x\):

1.

\[
x = \frac{72 + 68}{10} = \frac{140}{10} = 14.
\]

2.

\[
x = \frac{72 - 68}{10} = \frac{4}{10} = 0.4.
\]

Thus, the solutions to the equation \(-15x^2 + 216x - 84 = 0\) are:

\[
\boxed{14} \quad \text{and} \quad \boxed{0.4}.
\]