To factorize the cubic polynomial \( t^3 + 5t^2 + 8t + 40 \), we can use the method of finding one root and then dividing the polynomial by the corresponding linear factor. Let's go through the steps.

### Step 1: Find a Root
As previously calculated, one root of the polynomial is \( t = -5 \). Since \( t = -5 \) is a root, \( t + 5 \) is a factor of the polynomial.

### Step 2: Perform Polynomial Division
Now, we divide \( t^3 + 5t^2 + 8t + 40 \) by \( t + 5 \) to find the other factor.

We'll use synthetic division for this purpose:

```
-5 |  1   5   8   40
   |      -5   0  -40
   -------------------
       1   0   8   0
```

The quotient is \( t^2 + 8 \) and the remainder is 0, confirming that:

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

### Step 3: Express the Final Factorization
The polynomial \( t^2 + 8 \) can be factored further into:

\[
t^2 + 8 = t^2 + 4 \times 2 = (t + 2i\sqrt{2})(t - 2i\sqrt{2})
\]

However, if we factor over the reals, the factorization is complete at:

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

So, the final factorization over the real numbers is:

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

Would you like any further explanation or details?

Here are some related questions:

1. How do you perform synthetic division step-by-step?
2. What is the difference between factoring over the real numbers and complex numbers?
3. How do you factor quadratic expressions that don't have real roots?
4. What other methods can be used to factor cubic polynomials?
5. How do you check if a polynomial is factorable using the discriminant?

**Tip:** Always test simple roots like \( t = \pm 1, \pm 2, \pm 3 \) first when trying to factorize cubic polynomials, as they often simplify the factorization process.

Learn how to factorize the cubic polynomial t^3 + 5t^2 + 8t + 40 using synthetic division and polynomial factorization methods. This problem covers advanced algebraic concepts in polynomial factorization. Suitable for advanced high school students (Grades 11-12).

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