Final answer:
To solve the equation 4x⁴ + 12x³ + 5x² = 0 algebraically, we can factor out the common factor x² to get x²(4x² + 12x + 5) = 0. Then, we can solve each factor separately: x² = 0 (x = 0) and 4x² + 12x + 5 = 0 (x ≈ -0.5, -2).
Step-by-step explanation:
To solve the equation 4x⁴ + 12x³ + 5x² = 0 algebraically, we can factor out the common factor x² to get x²(4x² + 12x + 5) = 0. Then, we can solve each factor separately:
x² = 0: The only solution is x = 0.
4x² + 12x + 5 = 0: This quadratic equation can be solved using factoring, completing the square, or the quadratic formula. The solutions are approximately x = -0.5 and x = -2.
Therefore, the solutions to the equation are x = 0, x = -0.5, and x = -2.