Let's solve the equation \(x + \frac{3}{8} = \frac{2}{x+3}\).

1. **Isolate Fractions**:
Move terms around to get rid of the fractions. Start by equating the denominators or cross-multiply to eliminate the fractions. Multiply every term by \(8(x+3)\):

\[
8(x+3) \left( x + \frac{3}{8} \right) = 8(x+3) \cdot \frac{2}{x+3}
\]

After simplifying, we have:

\[
8(x(x+3)) + 3(x+3) = 16
\]

2. **Simplify**:
Distribute and simplify the equation:

\[
8x^2 + 24x + 3x + 9 = 16
\]

Combine like terms:

\[
8x^2 + 27x + 9 = 16
\]

3. **Move all terms to one side**:

\[
8x^2 + 27x + 9 - 16 = 0
\]

Simplify further:

\[
8x^2 + 27x - 7 = 0
\]

4. **Solve the Quadratic Equation**:
We can now use the quadratic formula, where \(a = 8\), \(b = 27\), and \(c = -7\):

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

Calculate the discriminant:

\[
b^2 - 4ac = 27^2 - 4 \cdot 8 \cdot (-7)
\]

\[
= 729 + 224 = 953
\]

So,

\[
x = \frac{-27 \pm \sqrt{953}}{16}
\]

5. **Conclusion**:
The solutions for the quadratic equation can be approximated by evaluating the square root term to find exact numerical values. However, they remain as theoretical solutions unless further simplified or evaluated numerically for specific needs. The two possible solutions arise from considering both the plus and minus options in the quadratic formula.