Let's determine $p$ such that

$$
x^{2}-4 b x+4 a b+p^{2}-2 p x
$$

is a perfect square.
The given expression can also be written as:

$$
x^{2}-2(2 b+p) x+\left(4 a b+p^{2}\right)
$$

This expression is a perfect square if

$$
(2 b+p)^{2}=4 a b+p^{2}
$$

from which

$$
p=a-b
$$

In this case, the given expression transforms into:

$$
x^{2}-2(a+b) x+(a+b)^{2}=[x-(a+b)]^{2}
$$

Number of solutions: 52.