Can You Prove It?

$a+b=12$ $\implies$ $a=12-b$ $\color{#20A900}(1)$

$ab=32$ $\color{#20A900}(2)$

$\text{Substitute}$ $\color{#20A900}(1)$ $\text{in}$ $\color{#20A900}(2)$

$(12-b)(b) = 32$ $\implies$ $\boxed{b^2 - 12b+32=0}$ $\text{by factoring, we get}$

$b = 8$

$b = 4$

$\text{Based from the values of}$ $b,$ $a$ $\text{is either}$ $8$ $\text{or}$ $4$ $\text{, but it says in the problem that}$ $a>b$ $\text{, so}$ $a = 8$ $\text{and}$ $b = 4$ .

$(a+2b)^2 - 5b^2 = (8+8)^2-5(4^2) =$ $\boxed{\large \color{#D61F06}176}$