1. I read your equation as: S = 14a^2+3b^2+12a+6b+3

2. If this is correct, then complete the squares at the RHS:

S = 14a^2+3b^2+12a+6b+3 ===> S = 14(a+3/7 )^2+3(b+1)^2-18/7

3. The smallest value of a square is zero. Therefore the smallest value of s occurs if both squares have the value zero. Determine a, b and s.