The given equation is:

$\sqrt{2x + 3} - 2 = x$

We need to transform this equation into a quadratic form.

Step 1: Isolate the square root term.

$\sqrt{2x + 3} = x + 2$

Step 2: Square both sides to eliminate the square root.

$(\sqrt{2x + 3})^2 = (x + 2)^2$ $2x + 3 = (x + 2)^2$

Step 3: Expand the right-hand side.

$2x + 3 = x^2 + 4x + 4$

Step 4: Bring all terms to one side of the equation.

$2x + 3 - (x^2 + 4x + 4) = 0$ $2x + 3 - x^2 - 4x - 4 = 0$

Step 5: Simplify the equation.

$-x^2 - 2x - 1 = 0$

Multiply by $-1$ to simplify further: $x^2 + 2x + 1 = 0$

Conclusion:

The quadratic equation is:

$x^2 + 2x + 1 = 0$

This matches option b: $x^2 + 2x + 1 = 0$.

Would you like any further details on this solution?

Here are five related questions to deepen your understanding:

1. How do we handle square root equations that involve more complex expressions?
2. What is the general process of transforming a square root equation into a quadratic equation?
3. What happens when squaring both sides of an equation introduces extraneous solutions?
4. How can you verify if your quadratic equation has real or complex solutions?
5. What are the conditions for a quadratic equation to have repeated roots?

Tip: Always check for extraneous solutions when squaring both sides of an equation.