1. Stating the problem: Simplify the expression $$C = \sqrt{x^2 + 2x + 1} - \sqrt{x^2 - 2x + c}$$ where $c$ is a constant. 2. Recognize that $x^2 + 2x + 1$ is a perfect square trinomial. It can be factored as: $$x^2 + 2x + 1 = (x+1)^2$$ 3. Rewrite the expression using this factorization: $$C = \sqrt{(x+1)^2} - \sqrt{x^2 - 2x + c}$$ 4. The square root of a square is the absolute value, so: $$\sqrt{(x+1)^2} = |x+1|$$ 5. The expression becomes: $$C = |x+1| - \sqrt{x^2 - 2x + c}$$ 6. To simplify further, consider the second square root. If $c=1$, then: $$x^2 - 2x + 1 = (x-1)^2$$ 7. Assuming $c=1$, the expression becomes: $$C = |x+1| - |x-1|$$ 8. The simplified form depends on the values of $x$ because of the absolute values: - If $x \geq 1$, then $|x+1| = x+1$ and $|x-1| = x-1$, so: $$C = (x+1) - (x-1) = 2$$ - If $-1 \leq x < 1$, then $|x+1| = x+1$ and $|x-1| = 1 - x$, so: $$C = (x+1) - (1 - x) = 2x$$ - If $x < -1$, then $|x+1| = -(x+1)$ and $|x-1| = 1 - x$, so: $$C = -(x+1) - (1 - x) = -2$$ Final answer: $$C = \begin{cases} 2 & x \geq 1 \\ 2x & -1 \leq x < 1 \\ -2 & x < -1 \end{cases}$$ If $c \neq 1$, the expression cannot be simplified further without knowing $c$.