We are given the system of equations:

$1. \ a + \sqrt{a} \times \sqrt{b} + b = 2$ $2. \ a^2 + ab + b^2 = 4$

Step 1: Simplify the first equation

The first equation involves square roots. We can simplify $\sqrt{a} \times \sqrt{b}$ to $\sqrt{ab}$. Thus, the first equation becomes:

$a + \sqrt{ab} + b = 2$

Step 2: Work with the second equation

The second equation is a quadratic form:

$a^2 + ab + b^2 = 4$

Step 3: Solve by trial or substitution

We will try some simple integer values of $a$ and $b$ to see if we can solve the system.

Trying $a = 1$ and $b = 1$:

For the first equation: $1 + \sqrt{1 \times 1} + 1 = 1 + 1 + 1 = 3 \neq 2$ This doesn't work.

Trying $a = 1$ and $b = 0$:

For the first equation: $1 + \sqrt{1 \times 0} + 0 = 1 + 0 + 0 = 1 \neq 2$ This doesn't work.

Trying $a = 1$ and $b = 1$:

Now, let's test (a = 2, b = -2.

In method