Final answer:
The question involves solving an unusual non-linear differential equation, which would typically require methods like separating variables or finding an integrating factor. For quadratic equations like t² + 10t - 200 = 0, one uses the quadratic formula.
Step-by-step explanation:
The student asked to determine the general solution of the differential equation y⁵ - 8 y' = t² + 6t + 4 + eᵗ. This equation is non-linear and unusual for standard solution methods; however, let's break down a possible approach to such problems generally. First, you would aim to separate variables or find an integrating factor, which would not be straightforward in this case. If the differential equation was mistyped and meant to be a more common form, such as y' + 8y = t² + 6t + 4 + et, then you could use methods for solving linear differential equations with non-homogeneous terms.
In solving a quadratic equation, for example, t² + 10t - 200 = 0, you'd use the quadratic formula: t = (-b ± √(b²-4ac))/(2a). This highlights the steps for solving algebraic equations where you rearrange the equation to get zero on one side and then apply the formula to find the values of t.