Unravel the Equation: Solving √(x+1) × √(x+2) = x+3

To solve the equation $\sqrt{x+1} \times \sqrt{x+2} = x+3$, we will follow these steps:

• Step 1: Identify the terms. Let $a = \sqrt{x+1}$ and $b = \sqrt{x+2}$, so $a \cdot b = x + 3$.
• Step 2: Square both sides of the equation to remove the square roots: $(ab)^2 = (x+3)^2$.
• Step 3: Express using the original variables: $(x+1)(x+2) = (x+3)^2$.
• Step 4: Expand both sides:
  Left side: $x^2 + 3x + 2$
  Right side: $x^2 + 6x + 9$.
• Step 5: Rearrange and simplify the equation by subtracting one side from the other:
  $x^2 + 3x + 2 = x^2 + 6x + 9$ becomes $0 = 3x + 7$.
• Step 6: Solve the linear equation: $3x = -7$, hence $x = -\frac{7}{3}$.
• Step 7: Verify that the solution $x = -\frac{7}{3}$ fits the initial domain requirements for the square roots.

Therefore, the solution to the problem is $x = -\frac{7}{3}$. This matches choice 3 in the provided answer choices.