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To find the vertical asymptotes of the function:
\[ f(x) = \frac{x \cdot (2x^2 - 58x - 60)}{k(x)} \]
where \( k(x) = x^3 - 41x^2 + 399x + 441 \), we need to find the values of \( x \) that make the denominator zero, provided these values do not also make the numerator zero.

### Step 1: Determine the roots of \( k(x) \)

Given \( k(x) = x^3 - 41x^2 + 399x + 441 \), and it has a root at \( x = -1 \). We can use synthetic division to factor \( k(x) \).

#### Synthetic Division
Dividing \( k(x) \) by \( x + 1 \):

```
-1 |  1   -41   399   441
   |      -1    42  -441
   ----------------------
      1   -42   441     0
```

This gives us:
\[ k(x) = (x + 1)(x^2 - 42x + 441) \]

### Step 2: Factorize the quadratic
\[ x^2 - 42x + 441 = (x - 21)^2 \]

Thus,
\[ k(x) = (x + 1)(x - 21)^2 \]

### Step 3: Find the roots of \( k(x) \)
The roots of \( k(x) \) are:
\[ x = -1 \]
\[ x = 21 \]

### Step 4: Simplify the numerator
\[ \text{Numerator} = x \cdot (2x^2 - 58x - 60) \]
\[ 2x^2 - 58x - 60 = 2(x^2 - 29x - 30) = 2(x - 30)(x + 1) \]

Thus, the numerator is:
\[ x \cdot 2(x - 30)(x + 1) = 2x(x - 30)(x + 1) \]

### Step 5: Simplify the function \( f(x) \)
\[ f(x) = \frac{2x(x - 30)(x + 1)}{(x + 1)(x - 21)^2} \]

Cancel the common factor \( (x + 1) \):
\[ f(x) = \frac{2x(x - 30)}{(x - 21)^2} \]

### Step 6: Determine the vertical asymptotes
The vertical asymptotes occur where the denominator is zero and the numerator is non-zero:
\[ (x - 21)^2 = 0 \]
Thus, \( x = 21 \).

### Answer:
The equation for the vertical asymptote of \( f(x) \) is:
\[ \boxed{x = 21} \]

If you have any further questions or need more details, feel free to ask! Here are some questions you might be interested in:

1. How do you perform synthetic division?
2. Can you explain how to factor polynomials?
3. What is the significance of vertical asymptotes in a function?
4. How can you verify the vertical asymptotes graphically?
5. Are there alternative methods to find vertical asymptotes?

**Tip:** Always check for common factors in the numerator and denominator when determining vertical asymptotes to simplify the function before analyzing it.

Learn how to find the vertical asymptotes of a rational function step-by-step. This problem involves polynomial division, factorization of polynomials, and identifying vertical asymptotes. Suitable for advanced high school students.

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