To solve the equation 3(x+2)=16(6-x)+1, we will first simplify both sides of the equation.

Starting with the left side:
3(x+2) = 3x + 6

Now let's simplify the right side:
16(6-x) + 1 = 96 - 16x + 1 = 97 - 16x

Now our equation becomes:
3x + 6 = 97 - 16x

Next, we will combine like terms by adding 16x to both sides:
3x + 16x + 6 = 97 - 16x + 16x
19x + 6 = 97

To isolate the variable x, we will subtract 6 from both sides:
19x + 6 - 6 = 97 - 6
19x = 91

Finally, we will divide both sides by 19 to solve for x:
19x/19 = 91/19
x = 91/19

Therefore, the solution to the equation 3(x+2)=16(6-x)+1 is x = 91/19.