To solve the equation \( 5x(69 - x) = 20 \), follow these steps:

1. Distribute the \( 5x \) across the terms inside the parentheses:
\[
5x \cdot 69 - 5x \cdot x = 20
\]
This simplifies to:
\[
345x - 5x^2 = 20
\]

2. Rearrange the equation to standard quadratic form \( ax^2 + bx + c = 0 \):
\[
-5x^2 + 345x - 20 = 0
\]
Multiply through by \(-1\) to make the coefficient of \( x^2 \) positive:
\[
5x^2 - 345x + 20 = 0
\]

3. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 5 \), \( b = -345 \), and \( c = 20 \):
\[
x = \frac{-(-345) \pm \sqrt{(-345)^2 - 4 \cdot 5 \cdot 20}}{2 \cdot 5}
\]
Simplify inside the square root:
\[
x = \frac{345 \pm \sqrt{119025 - 400}}{10}
\]
\[
x = \frac{345 \pm \sqrt{118625}}{10}
\]

4. Calculate the square root of 118625:
\[
\sqrt{118625} \approx 344.52
\]

5. Substitute back into the quadratic formula:
\[
x = \frac{345 \pm 344.52}{10}
\]

6. Solve for the two possible values of \( x \):
\[
x = \frac{345 + 344.52}{10} = \frac{689.52}{10} = 68.952
\]
\[
x = \frac{345 - 344.52}{10} = \frac{0.48}{10} = 0.048
\]

So, the solutions to the equation \( 5x(69 - x) = 20 \) are:
\[
x \approx 68.952 \quad \text{and} \quad x \approx 0.048
\]