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To solve the inequality:

\[
(x + 6)(x^2 + 16)(1 - x) \leq 0
\]

we need to analyze the signs of each factor and find the intervals where their product is less than or equal to zero.

### Step 1: Analyze Each Factor

1. **First Factor: \(x + 6\)**

   - This factor is zero when \(x = -6\).
   - For \(x < -6\), \(x + 6 < 0\).
   - For \(x > -6\), \(x + 6 > 0\).

2. **Second Factor: \(x^2 + 16\)**

   - This expression is a sum of squares, so \(x^2 + 16 \geq 16\) for all real \(x\).
   - It is always positive, \(x^2 + 16 > 0\), for all \(x\).

3. **Third Factor: \(1 - x\)**

   - This factor is zero when \(x = 1\).
   - For \(x < 1\), \(1 - x > 0\).
   - For \(x > 1\), \(1 - x < 0\).

### Step 2: Determine the Intervals for the Product

We now consider the product \((x + 6)(x^2 + 16)(1 - x) \leq 0\). Since \(x^2 + 16 > 0\) for all real \(x\), it does not affect the sign of the product. The sign of the product is determined by the factors \(x + 6\) and \(1 - x\). 

We analyze the intervals divided by the critical points \(x = -6\) and \(x = 1\):

1. **Interval 1: \(x < -6\)**
   - \(x + 6 < 0\)
   - \(1 - x > 0\)
   - Product: Negative

2. **Interval 2: \(-6 < x < 1\)**
   - \(x + 6 > 0\)
   - \(1 - x > 0\)
   - Product: Positive

3. **Interval 3: \(x > 1\)**
   - \(x + 6 > 0\)
   - \(1 - x < 0\)
   - Product: Negative

### Step 3: Include Points Where the Product is Zero

The product is zero at \(x = -6\) and \(x = 1\).

### Step 4: Write the Solution in Interval Notation

The inequality is satisfied when the product is less than or equal to zero. This happens in:

- Interval \((-\infty, -6]\) where the product is negative.
- Point \(x = -6\) where the product is zero.
- Interval \([1, \infty)\) where the product is negative.

### Final Solution

\[
(-\infty, -6] \cup [1, \infty)
\]

If you have any questions or need further details, please let me know!

### Related Questions

1. How would the solution change if the inequality were strictly less than zero (\(< 0\))?
2. What happens if one of the factors was negative or a more complex expression?
3. How do you solve inequalities with rational expressions?
4. Can this inequality be represented graphically on a number line? What would it look like?
5. What if we had \(x^2 - 16\) instead of \(x^2 + 16\)? How would the intervals change?

### Tip

Always remember to check the points where each factor becomes zero; these are crucial in determining the intervals for inequalities.

(x + 6)(x^2 + 16)(1 - x) ≤ 0. Write your answer as an interval or union of intervals.

Learn how to solve the polynomial inequality (x + 6)(x^2 + 16)(1 - x) ≤ 0 with a detailed step-by-step solution. This problem covers key algebraic concepts like product of factors, interval notation, and sign analysis. Suitable for high school students in Grades 10-12.

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