Step-by-Step Solution:

1. Understand the Complex Number Condition: The given condition is arg((z + α) / (z + β)) = π / 4. Using the property of arguments, arg(w₁ / w₂) = arg(w₁) - arg(w₂), we can rewrite the condition as: $arg(z + α) - arg(z + β) = {π \over 4}$

2. Convert to Cartesian Coordinates: Let the complex number z be represented by the point (x, y), so z = x + iy. Since α and β are real numbers, we have: z + α = (x + α) + iy z + β = (x + β) + iy

   Let θ₁ = arg(z + α) and θ₂ = arg(z + β). The condition is θ₁ - θ₂ = π / 4. In terms of x and y: tan(θ₁) = y / (x + α) tan(θ₂) = y / (x + β)

3. Derive the Locus Equation: We take the tangent of the argument difference: $tan(θ₁ - θ₂) = tan(π / 4) = 1$ Using the tangent subtraction formula, tan(A - B) = (tan A - tan B) / (1 + tan A tan B): ${{tan(θ₁) - tan(θ₂)} \over {1 + tan(θ₁)tan(θ₂)}} = 1$ {{y \over {x + α}} - {y \over {x + β}}} \over {1 + \left( {y \over {x + α}} \right)\left( {y \over {x + β}} \right)}} = 1 Simplifying the numerator and denominator: ${{y(x + β) - y(x + α)} \over {(x + α)(x + β)}} \over {{(x + α)(x + β) + y^2} \over {(x + α)(x + β)}} = 1$ This simplifies to: $y((x + β) - (x + α)) = (x + α)(x + β) + y^2$ $y(β - α) = x^2 + (α + β)x + αβ + y^2$ Rearranging the terms, we get the equation of a circle: $x^2 + y^2 + (α + β)x - (β - α)y + αβ = 0$

4. Compare with the Given Circle Equation: The problem states that the ordered pair (x, y) lies on the circle x² + y² + 5x - 3y + 4 = 0. This means the locus we derived is identical to the given circle. We can compare the coefficients of the two equations: $x^2 + y^2 + (α + β)x - (β - α)y + αβ = 0$ $x^2 + y^2 + 5x - 3y + 4 = 0$

   Comparing the coefficients, we get a system of equations:

   1. α + β = 5
   2. -(β - α) = -3 => β - α = 3
   3. αβ = 4

5. Solve for α and β: We have a system of two linear equations for α and β: α + β = 5 β - α = 3 Adding these two equations gives: 2β = 8 => β = 4 Substituting β = 4 into the first equation: α + 4 = 5 => α = 1

6. Verify the Solution: Let's check if these values satisfy the third equation αβ = 4: (1)(4) = 4. The values are consistent. So, we have uniquely determined α = 1 and β = 4.

7. Evaluate the Options: Now we check each statement with the values α = 1 and β = 4.

   • A: α = -1: This is FALSE because α = 1.
   • B: αβ = 4: This is TRUE because (1)(4) = 4.
   • C: αβ = -4: This is FALSE.
   • D: β = 4: This is TRUE.

Conclusion:

The correct statements are B and D.