Consider the equation |x² - 2x - 3| = m. m belongs to ℝ. If the given equation has four solutions, then

• m ∈ (0, ∞)
• m ∈ (-1, 3)
• m ∈ (0, 4)
• None of these

To determine the values of \( m \) for which the equation \( |x² - 2x - 3| = m \) has four solutions, we first analyze the expression inside the absolute value.

Factoring the Quadratic

The quadratic \( x² - 2x - 3 \) can be factored as:

• \( (x - 3)(x + 1) \)

This means the roots of the equation \( x² - 2x - 3 = 0 \) are \( x = 3 \) and \( x = -1 \).

Behavior of the Quadratic

The parabola opens upwards (since the coefficient of \( x² \) is positive) and has its vertex at \( x = 1 \). Evaluating the quadratic at the vertex:

• \( f(1) = 1² - 2(1) - 3 = -4 \)

This indicates that the minimum value of \( x² - 2x - 3 \) is \(-4\), occurring at \( x = 1 \).

Analyzing the Absolute Value

The equation \( |x² - 2x - 3| = m \) will have four solutions when the graph of \( y = x² - 2x - 3 \) intersects the line \( y = m \) at two points above and below the x-axis. This occurs when:

• The line \( y = m \) is above the minimum value of the quadratic (which is \(-4\)) and below the x-axis (which is \(0\)).

Finding the Range for m

Thus, for four solutions, \( m \) must satisfy:

• \( m > 0 \) (to be above the x-axis)
• \( m < 4 \) (to be below the vertex value of the parabola)

Combining these conditions, we find that \( m \) must be in the interval \( (0, 4) \).

Final Answer

Therefore, the correct option is:

• m ∈ (0, 4)