Recall that x+x1≥2 for all positive real numbers x, with equality if and only if x=1. Therefore the left-hand side of this equation is bounded below by 8+8+8+8, and the only solution occurs when 2a=1 and b=1.
OR
Applying the Arithmetic Mean-Geometric Mean Inequality gives the following statements:
1+2a=21+21+a+a≥444a2; equality holds if and only if a=21;
2+2b=1+1+b+b≥44b2; equality holds if and only if b=1;
2a+b=a+a+2b+2b≥444a2b2; equality holds if and only if b=2a.
Multiplying the three inequalities above gives (1+2a)(2+2b)(2a+b)≥32ab, and equality holds if and only if a=21 and b=1.