Advanced Inequalities

$a+\sqrt{b}$, its conjugate: $a-\sqrt{b}$

If their sum is -4:

$a+\sqrt{b}+a-\sqrt{b}=-4 \implies 2a=-4 \implies a=-2$

If their product is 1:

$(2+\sqrt{b})(2-\sqrt{b})=1 \iff 4-2\sqrt{b}+2\sqrt{b}-b=1$

So:

$4-b=1 \implies b=3$

Therefore, $a+b=-2+3=1$