Find the third proportional to a² - b² and a + b.

Let third proportional to a² - b² and a + b be x,

$\Rightarrow \dfrac{a^2 -b^2}{a + b} = \dfrac{a + b}{x} \\[1em] \Rightarrow x = \dfrac{(a + b)(a + b)}{a^2- b^2} \\[1em] \Rightarrow x = \dfrac{(a + b)(a + b)}{(a + b)(a - b)} \\[1em] \Rightarrow x = \dfrac{(a + b)}{(a - b)}.$

Hence, third proportional to a² - b² and a + b = $\dfrac{(a + b)}{(a - b)}.$