What is the mean proportional ratio between $(3+\sqrt{5})$ and $15-\sqrt{125}$?

The mean proportional between two expressions $a$ and $b$ is given by $\sqrt{ab}$.Here:
$$a=3+\sqrt{5}$$
$$b=15-\sqrt{125}=15-\sqrt{25\times 5}=15-5\sqrt{5}$$
We can factor out $5$ from the expression for $b$:
$$b=5(3-\sqrt{5})$$
Now, find the product $ab$:
$$ab=(3+\sqrt{5})\times 5(3-\sqrt{5})$$
$$ab=5\times (3+\sqrt{5})(3-\sqrt{5})$$
Using the identity $(x+y)(x-y)=x^2-y^2$:
$$(3+\sqrt{5})(3-\sqrt{5})=3^2-(\sqrt{5}{)}^2=9-5=4$$
So,
$$ab=5\times 4=20$$
The mean proportional is:
$$\sqrt{ab}=\sqrt{20}=\sqrt{4\times 5}=2\sqrt{5}$$