To find the value of \( x \), we start by noting that the two adjacent angles are inside a 90-degree angle. This means that the sum of these two angles must equal 90 degrees.
The angles are given as \( x + 4 \) and \( 3x + 2 \). Therefore, we can set up the following equation:
\[ (x + 4) + (3x + 2) = 90 \]
Now, we simplify the left side:
\[ x + 4 + 3x + 2 = 90 \]
Combine like terms:
\[ 4x + 6 = 90 \]
Next, we subtract 6 from both sides:
\[ 4x = 90 - 6 \] \[ 4x = 84 \]
Now, divide both sides by 4:
\[ x = \frac{84}{4} = 21 \]
Now that we have the value of \( x \), we can conclude:
\[ \boxed{21} \]