Algebraic Identity Evaluation

Given $x^2 = y + z$. Add $x$ on both sides: $x^2 + x = x + y + z \implies x(x+1) = x + y + z \implies \frac{1}{x+1} = \frac{x}{x+y+z}$

Similarly: $\frac{1}{y+1} = \frac{y}{x+y+z}, \quad \frac{1}{z+1} = \frac{z}{x+y+z}$

Adding these three terms: $\frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} = \frac{x+y+z}{x+y+z} = 1$