The equation you provided is:

$a \sqrt{x + 1} = b$

We need to solve for $x$. Here's the step-by-step process:

1. Divide both sides by $a$: $\sqrt{x + 1} = \frac{b}{a}$

2. Square both sides to eliminate the square root: $x + 1 = \left(\frac{b}{a}\right)^2$

3. Subtract 1 from both sides: $x = \left(\frac{b}{a}\right)^2 - 1$

Thus, the solution is: $x = \left(\frac{b}{a}\right)^2 - 1$

Let me know if you'd like more details or have any questions.

Here are 5 related questions to explore:

1. What are the conditions on $a$ and $b$ for the solution to be valid?
2. How would you solve if there were a constant added to $b$ on the right-hand side?
3. What if the square root was on both sides of the equation?
4. How would this solution change if $x + 1$ were cubed instead of square rooted?
5. How do we interpret the geometric meaning of square roots in equations?

Tip: Always be cautious when squaring both sides of an equation, as it can introduce extraneous solutions. Always verify your solution by substituting back into the original equation.