To solve this equation, we can start by squaring both sides to eliminate the square root:

((x+1)√(x^2-2x+3))^2 = (x^2+1)^2

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

Expanding both sides:

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

x^4 - 4x^2 + 3 + 2x^3 - 4x + 2x^2 - 2x - 4 + 3x^2 - 6x + 9 = x^4 + 2x^2 + 1

Combining like terms:

x^4 + 2x^3 - 4x + 3x^2 - 6x + 9 - 4x^2 - 4 = x^4 + 2x^2 + 1

Simplifying:

x^4 + 2x^3 - 4x + 3x^2 - 6x + 9 - 4x^2 - 4 - x^4 - 2x^2 - 1 = 0

x^3 - 7x + 4 = 0

Unfortunately, this equation does not have a simple algebraic solution. It can be solved numerically using methods such as graphing or using a calculator.