To solve the equation \( 5x^2 + 8x - 69 = 0 \) by factoring, we first need to find two numbers that multiply to \( a \cdot c = 5 \cdot (-69) = -345 \) and add to \( b = 8 \).
We are looking for two numbers that multiply to \(-345\) and add to \(8\). After considering various factor pairs, we find:
- The numbers are \( 15 \) and \(-23\) because \( 15 \cdot (-23) = -345 \) and \( 15 + (-23) = -8\).
Next, we can rewrite the middle term using these numbers:
\[ 5x^2 + 15x - 23x - 69 = 0 \]
Now, we can factor by grouping:
\[ (5x^2 + 15x) + (-23x - 69) = 0 \] \[ 5x(x + 3) - 23(x + 3) = 0 \]
Now factor out the common term \((x + 3)\):
\[ (5x - 23)(x + 3) = 0 \]
Now we can set each factor equal to zero:
- \(5x - 23 = 0\)
- \(x + 3 = 0\)
Solving for \(x\) in the first equation:
\[ 5x = 23 \] \[ x = \frac{23}{5} \]
Solving for \(x\) in the second equation:
\[ x = -3 \]
The solutions are:
\[ x = \frac{23}{5}, -3 \]
Thus, the final answers are:
\[ \boxed{\frac{23}{5}, -3} \]