Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

The given equation is $x^{2}-2 a x-\left(4 b^{2}-a^{2}\right)=0$.

Comparing it with $A x^{2}+B x+C=0$, we get

$A=1, B=-2 a$ and $C=-\left(4 b^{2}-a^{2}\right)$

$\therefore$ Discriminant, $D=B^{2}-4 A C=(-2 a)^{2}-4 \times 1 \times\left[-\left(4 b^{2}-a^{2}\right)\right]=4 a^{2}+16 b^{2}-4 a^{2}=16 b^{2}>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{16 b^{2}}=4 b$

$\therefore \alpha=\frac{-B+\sqrt{D}}{2 A}=\frac{-(-2 a)+4 b}{2 \times 1}=\frac{2(a+2 b)}{2}=a+2 b$

$\beta=\frac{-B-\sqrt{D}}{2 A}=\frac{-(-2 a)-4 b}{2 \times 1}=\frac{2(a-2 b)}{2}=a-2 b$

Hence, $a+2 b$ and $a-2 b$ are the roots of the given equation.