From $x=a^2-2$ and $y=b^2-2$, rewrite $x+y=14$ as an equation involving $a^2+b^2$.

Square $a-b=1$ to create $a^2+b^2-2ab=1$, then use your value of $a^2+b^2$ to find $ab$.

Write $xy=(a^2-2)(b^2-2)$ and expand. Replace $a^2b^2$ with $(ab)^2$ and replace $a^2+b^2$ with the value you found.

Graph the line $y=14-x$.

Then graph the equation $\sqrt{x+2}-\sqrt{y+2}=1$ (you can type it exactly as written; Desmos will graph it as an implicit curve).

Click on the intersection of the two graphs and create a point (for example, point $A$). P‍ro‌pе⁠rty оf⁠ Aniko.‍a⁠i

In a new expression line, type A.x*A.y.

Match that value to the answer choice.

Let $a=\sqrt{x+2}$ and $b=\sqrt{y+2}$. Then $a\ge0$ and $b\ge0$, and the system becomes

From $x+y=14$,

so

Square the equation $a-b=1$: Wrіtten‍ b‍y‍ A‌nікo

Substitute $a^2+b^2=18$:

Since $x=a^2-2$ and $y=b^2-2$,

Use $a^2b^2=(ab)^2$ and $a^2+b^2=18$:

Substitute $ab=\frac{17}{2}$ and $a^2+b^2=18$:

So the correct choice is $\frac{161}{4}$.