To solve the equation
\[ \frac{x + 8}{5} = \frac{2}{5x - 6}, \]
we can start by cross-multiplying to eliminate the fractions:
\[ (x + 8)(5x - 6) = 2 \cdot 5. \]
Expanding both sides, we get:
\[ x(5x - 6) + 8(5x - 6) = 10. \]
This simplifies to:
\[ 5x^2 - 6x + 40x - 48 = 10. \]
Combining like terms gives:
\[ 5x^2 + 34x - 48 = 10. \]
Now, we can move \(10\) to the left side:
\[ 5x^2 + 34x - 58 = 0. \]
Now, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 5\), \(b = 34\), and \(c = -58\).
First, we need to calculate the discriminant:
\[ b^2 - 4ac = 34^2 - 4 \cdot 5 \cdot (-58). \]
Calculating this gives:
\[ 1156 + 1160 = 2316. \]
Now, applying the quadratic formula:
\[ x = \frac{-34 \pm \sqrt{2316}}{2 \cdot 5}. \]
Calculating \(\sqrt{2316}\):
\[ \sqrt{2316} \approx 48.12. \]
So we have:
\[ x = \frac{-34 \pm 48.12}{10}. \]
This gives us two potential solutions:
- \(x = \frac{-34 + 48.12}{10} \approx \frac{14.12}{10} \approx 1.412\).
- \(x = \frac{-34 - 48.12}{10} \approx \frac{-82.12}{10} \approx -8.212\).
Thus, the solutions to the equation are approximately:
\[ x \approx 1.412 \quad \text{and} \quad x \approx -8.212. \]