Variation of Jun Shin's problem

The answer $10$ is incorrect because there is no pair of $(a,b)$ exists that gives $(a+1)(b+1) = 10$ . Using Langrange multipliers, we note that the expression is minimum when $a=b$ . Then $a^2(2a+1) = 25$ . Solving it, we get $a = b = 2.165526506$ , therefore $(a+1)(b+1)= 3.165526506^2 = \boxed{10.02055806}$ .

I have also solved it using graphing and get the same result. Solving $b = \dfrac{-a-1+\sqrt{(a+1)+4\times25/a}}{2}$ for different $a$ and then find the minimum $(a+1)(b+1)$ .