To solve the absolute value equation
\[ 13 - 4 |15 + 3x| + 3 = 16, \]
we first simplify the left side:
\[ 13 + 3 - 4 |15 + 3x| = 16, \]
which simplifies to
\[ 16 - 4 |15 + 3x| = 16. \]
Next, we subtract 16 from both sides:
\[ -4 |15 + 3x| = 0. \]
Dividing by -4, we find:
\[ |15 + 3x| = 0. \]
The absolute value of an expression is zero only when the expression itself is zero. Thus, we set the inside of the absolute value to zero:
\[ 15 + 3x = 0. \]
Solving for \(x\), we subtract 15 from both sides:
\[ 3x = -15, \]
and then divide by 3:
\[ x = -5. \]
Thus, the solution to the absolute value equation is
\[ \boxed{-5}. \]