X^2+3x-x√x^2+2=1+2√x^2+2

To solve this equation, we can start by simplifying both sides:

x^2 + 3x - x√(x^2 + 2) + 1 = 1 + 2√(x^2 + 2)

Next, we can subtract 1 from both sides to eliminate the constant term:

x^2 + 3x - x√(x^2 + 2) = 2√(x^2 + 2)

Now, let's isolate the square root term on one side of the equation:

x^2 + 3x - 2√(x^2 + 2) = x√(x^2 + 2)

Next, we can square both sides of the equation to eliminate the square root:

(x^2 + 3x - 2√(x^2 + 2))^2 = (x√(x^2 + 2))^2

Expanding both sides:

x^4 + 6x^3 + 4x^2 - 4x^2√(x^2 + 2) - 12x√(x^2 + 2) + 4(x^2 + 2) = x^4 + 2x^2

Simplifying:

6x^3 - 4x^2√(x^2 + 2) - 12x√(x^2 + 2) + 4x^2 + 8 - 2x^2 = 0

Combining like terms:

6x^3 - 2x^2 - 4x^2√(x^2 + 2) - 12x√(x^2 + 2) + 8 = 0

Now, we can try to factor out common terms:

2(3x^3 - x^2 - 2x^2√(x^2 + 2) - 6x√(x^2 + 2) + 4) = 0

At this point, it doesn't seem possible to factor out any common terms. Therefore, we can try to solve for x numerically using numerical methods or approximation techniques.