Final answer:
The equation \(\sqrt{2x + 1} + 4 = -1\) has no real solutions because it implies that the square root of a real number is negative, which is impossible. Isolating the square root term shows that the resulting value conflicts with the property that square roots of non-negative numbers are always non-negative.
Step-by-step explanation:
The equation \(\sqrt{2x + 1} + 4 = -1\) inherently suggests that \(\sqrt{2x + 1}\) should result in a negative number, since adding 4 and getting -1 would imply that the square root expression is equal to -5. However, this is impossible because the square root of any real number cannot be negative. The square root function always yields non-negative results for non-negative inputs. Therefore, there are no real solutions to the equation because it violates the fundamental property of square roots.
When attempting to solve, if we were to isolate the square root term, we would have \(\sqrt{2x + 1} = -5\). Squaring both sides to eliminate the square root gives us 2x + 1 = 25, which further simplifies to 2x = 24, thus x = 12. However, this solution is invalid since it contradicts the initial condition that the square root term cannot be negative. Checking back, substituting x = 12 into the original equation \(\sqrt{2x + 1}\) would result in a positive number, not -5. Hence, the equation has no real solutions.