1. **State the problem:** Simplify the expression $2(ab+3)^2 - 7(5 - ab)$. 2. **Recall formulas and rules:** Use the expansion formula for a square: $$ (x+y)^2 = x^2 + 2xy + y^2 $$ and distribute multiplication over addition/subtraction. 3. **Expand the square:** $$ (ab+3)^2 = (ab)^2 + 2 \cdot ab \cdot 3 + 3^2 = a^2b^2 + 6ab + 9 $$ 4. **Multiply by 2:** $$ 2(ab+3)^2 = 2(a^2b^2 + 6ab + 9) = 2a^2b^2 + 12ab + 18 $$ 5. **Distribute the -7:** $$ -7(5 - ab) = -7 \cdot 5 + (-7)(-ab) = -35 + 7ab $$ 6. **Combine all terms:** $$ 2a^2b^2 + 12ab + 18 - 35 + 7ab = 2a^2b^2 + (12ab + 7ab) + (18 - 35) $$ 7. **Simplify like terms:** $$ 2a^2b^2 + 19ab - 17 $$ **Final answer:** $$ \boxed{2a^2b^2 + 19ab - 17} $$