1. **State the problem:** We are given the expression $$\frac{x^2 - c}{x - b}$$ where $b$ and $c$ are positive integers. It is stated that this expression is equivalent to $$x + b$$ and that $$x \neq b$$. 2. **Understand the problem:** Since the expression $$\frac{x^2 - c}{x - b}$$ simplifies to $$x + b$$ for all $$x \neq b$$, it means that $$x^2 - c$$ is divisible by $$x - b$$ and the quotient is $$x + b$$. 3. **Use polynomial division or factorization:** If $$\frac{x^2 - c}{x - b} = x + b$$, then multiplying both sides by $$x - b$$ gives: $$ x^2 - c = (x - b)(x + b) $$ 4. **Expand the right side:** $$ (x - b)(x + b) = x^2 + bx - bx - b^2 = x^2 - b^2 $$ 5. **Equate the expressions:** $$ x^2 - c = x^2 - b^2 $$ 6. **Simplify:** $$ -c = -b^2 \implies c = b^2 $$ 7. **Interpretation:** The value of $$c$$ must be the square of $$b$$. 8. **Check the options:** - A) 4 = $2^2$ (possible if $b=2$) - B) 6 (not a perfect square) - C) 8 (not a perfect square) - D) 10 (not a perfect square) 9. **Conclusion:** The only possible value of $$c$$ from the options given is 4. **Final answer:** 4