1. **Stating the problem:** Given the function $$R(x) = (4 + a) \sqrt{2x + 1} - x + 3 - y$$ and the condition $$P(4) = 7$$, we want to find the value of $$a$$ assuming $$y$$ is a constant or zero. 2. **Understanding the function:** The function involves a square root term $$\sqrt{2x + 1}$$ multiplied by $$(4 + a)$$, then subtracting $$x$$, adding 3, and subtracting $$y$$. 3. **Using the condition $$P(4) = 7$$:** This means when $$x = 4$$, the function value is 7. 4. **Substitute $$x = 4$$ into $$R(x)$$:** $$R(4) = (4 + a) \sqrt{2(4) + 1} - 4 + 3 - y = 7$$ 5. **Simplify inside the square root:** $$\sqrt{8 + 1} = \sqrt{9} = 3$$ 6. **Rewrite the equation:** $$ (4 + a) \times 3 - 4 + 3 - y = 7 $$ 7. **Simplify constants:** $$ 3(4 + a) - 1 - y = 7 $$ 8. **Isolate terms:** $$ 3(4 + a) = 7 + 1 + y $$ $$ 3(4 + a) = 8 + y $$ 9. **Divide both sides by 3:** $$ \cancel{3}(4 + a) = \frac{8 + y}{\cancel{3}} $$ 10. **Solve for $$a$$:** $$ 4 + a = \frac{8 + y}{3} $$ $$ a = \frac{8 + y}{3} - 4 $$ **Final answer:** $$ a = \frac{8 + y}{3} - 4 $$ This expresses $$a$$ in terms of $$y$$. If $$y$$ is known or zero, substitute to find $$a$$.