Question:
Let $a, b, c$ be distinct complex numbers such that \[\begin{aligned} 2a^3 + 7a^2 - 8a + 5 &= 0, \\ 2b^3 + 7b^2 - 8b + 5 &= 0, \\ 2c^3 + 7c^2 - 8c + 5 &= 0. \end{aligned}\]Compute the value of the product $abc.$

Answer:
We are given that the roots of the polynomial equation \[2x^3 + 7x^2 - 8x + 5 = 0\]are $a, b, c$ (because they all satisfy the equation). Therefore, by Vieta's formulas, $abc = \boxed{-\tfrac{5}{2}}.$