To solve the equation \(-40 = t^3 + 5t^2 + 8t\), we first rewrite it in standard form by moving \(-40\) to the other side of the equation:

\[
t^3 + 5t^2 + 8t + 40 = 0
\]

Now, we can try to solve this cubic equation by factoring, if possible, or using numerical methods such as the Rational Root Theorem or synthetic division. Let's first try to find possible rational roots.

### Step 1: Check for Rational Roots
Possible rational roots can be given by the factors of the constant term (40) divided by the factors of the leading coefficient (1). The factors of 40 are \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40\). Since the leading coefficient is 1, the possible rational roots are:

\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40
\]

### Step 2: Test the Possible Roots
We test each of these possible roots by substituting them into the equation \(t^3 + 5t^2 + 8t + 40 = 0\) until we find one that satisfies the equation.

Testing \( t = -4 \):

\[
(-4)^3 + 5(-4)^2 + 8(-4) + 40 = -64 + 80 - 32 + 40 = 24 \neq 0
\]

Testing \( t = -5 \):

\[
(-5)^3 + 5(-5)^2 + 8(-5) + 40 = -125 + 125 - 40 + 40 = 0
\]

Since \(t = -5\) is a root, we can factor the cubic polynomial as:

\[
t^3 + 5t^2 + 8t + 40 = (t + 5)(t^2 + 0t + 8)
\]

### Step 3: Solve the Quadratic Equation
The remaining factor is \(t^2 + 8 = 0\), which is a quadratic equation:

\[
t^2 + 8 = 0
\]

Solve for \(t\):

\[
t^2 = -8
\]
\[
t = \pm \sqrt{-8}
\]
\[
t = \pm 2\sqrt{2}i
\]

### Step 4: Combine All Solutions
The solutions for \(t\) are:

\[
t = -5, \quad t = 2\sqrt{2}i, \quad t = -2\sqrt{2}i
\]

So, the complete solution set is:

\[
t = -5, \, t = 2\sqrt{2}i, \, t = -2\sqrt{2}i
\]

Would you like more details on any part of this solution? 

Here are some related questions:

1. How do you perform synthetic division to solve cubic equations?
2. What are complex numbers and how do they appear in solutions?
3. What is the Rational Root Theorem and how is it applied?
4. How can we factor a cubic polynomial by grouping?
5. What are the properties of cubic equations in general?

**Tip:** When solving higher-degree polynomials, always look for possible rational roots using the Rational Root Theorem as your first step.

Explore how to solve the cubic equation t^3 + 5t^2 + 8t = -40 step-by-step. Understand the application of the Rational Root Theorem, factorization of cubic polynomials, and the introduction of complex numbers in solutions. Suitable for advanced high school students.

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