1. State the problem: We are given two functions $f(x) = x^2 + 1$ and $g(x) = x - 5$. We need to solve the equation $f(g(x)) = g(f(x))$. 2. Find $f(g(x))$: Substitute $g(x)$ into $f$. $$f(g(x)) = f(x - 5) = (x - 5)^2 + 1$$ Expand the square: $$= (x^2 - 10x + 25) + 1 = x^2 - 10x + 26$$ 3. Find $g(f(x))$: Substitute $f(x)$ into $g$. $$g(f(x)) = g(x^2 + 1) = (x^2 + 1) - 5 = x^2 - 4$$ 4. Set the two expressions equal and solve for $x$: $$x^2 - 10x + 26 = x^2 - 4$$ Subtract $x^2$ from both sides: $$-10x + 26 = -4$$ Add $10x$ to both sides: $$26 = 10x - 4$$ Add 4 to both sides: $$30 = 10x$$ Divide both sides by 10: $$x = 3$$ 5. Final answer: $$x = 3$$