To determine how many solutions the equation \( 4s + 8 = 3 + 5 + 4s \) has, we can simplify it.
First, let's simplify the right-hand side of the equation:
\[ 3 + 5 = 8 \]
So the equation becomes:
\[ 4s + 8 = 8 + 4s \]
Next, we can subtract \( 4s \) from both sides:
\[ 8 = 8 \]
This statement is always true. Since both sides of the equation are equal regardless of the value of \( s \), it means that there are infinitely many solutions.
Thus, the final answer is:
Infinitely many