Final answer:
The equation x⁴ + 2x³ - 28x² + 50x + 75 = 0 is a quartic and may require advanced methods or software to solve. The quadratic formula can apply if the equation reduces to quadratic factors.
Step-by-step explanation:
To find all solutions of the equation x⁴ + 2x³ - 28x² + 50x + 75 = 0, we could use factoring, the rational root theorem, synthetic division, or numerical methods, but since it's not easily factorable and it's a quartic (fourth-degree) equation, these methods may be quite complex. Instead, we will often turn to powerful algebra software or graphing calculators to find the roots.
However, it's also possible that the equation may be reduced to quadratic factors, in which case we could use the quadratic formula to find the roots. In general, for a quadratic equation of the form ax² + bx + c = 0, the solutions can be calculated using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are constants, and √ represents the square root function.