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We are given the function:

\[
f(x) = -3x(4x^3 - 1)(7x^2 + 1)(x + 1)
\]

We need to determine the end behavior of this polynomial function. The end behavior of a polynomial is primarily influenced by two things:
1. The degree of the polynomial.
2. The sign of the leading term (i.e., the coefficient of the highest power of \(x\)).

### Step 1: Determining the Degree
Each factor contributes to the degree of the polynomial:
- \(x\) contributes 1 (from \(x^1\)).
- \(4x^3 - 1\) contributes 3 (since the highest power is \(x^3\)).
- \(7x^2 + 1\) contributes 2 (since the highest power is \(x^2\)).
- \(x + 1\) contributes 1 (from \(x^1\)).

Thus, the total degree of the polynomial is:

\[
1 + 3 + 2 + 1 = 7
\]

This is a degree 7 polynomial.

### Step 2: Leading Coefficient
The leading term comes from multiplying the highest power of \(x\) from each factor:
\[
-3x \cdot 4x^3 \cdot 7x^2 \cdot x = -3 \cdot 4 \cdot 7 \cdot x^{1+3+2+1} = -84x^7
\]

The leading coefficient is \(-84\), and the degree is odd (7).

### Step 3: End Behavior Analysis
For polynomials, the end behavior depends on the degree and sign of the leading coefficient:
- **Odd degree and negative leading coefficient**: 
  - As \(x \to -\infty\) (left), \(f(x) \to \infty\).
  - As \(x \to \infty\) (right), \(f(x) \to -\infty\).

Thus, the function **increases to the left** and **decreases to the right**.

### Correct Answer:
\[
\text{increasing to the left and decreasing to the right}
\]

Do you want more details or have any other questions?

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Here are 5 related questions for further exploration:
1. How does the degree of a polynomial affect its end behavior?
2. What is the impact of changing the sign of the leading coefficient on the graph of a polynomial?
3. Can you find the x-intercepts of the function \(f(x)\)?
4. How do you determine whether a polynomial has symmetry (even, odd, or neither)?
5. What is the behavior of the function near critical points (like where it equals 0)?

**Tip:** Always consider the degree and leading coefficient to predict the end behavior of polynomials.

Which set of words describes the end behavior of the function f(x) = -3x(4x^3 - 1)(7x^2 + 1)(x + 1)?

Learn how to analyze the end behavior of the polynomial function f(x) = -3x(4x^3 - 1)(7x^2 + 1)(x + 1). This problem covers key concepts such as determining the degree, leading coefficient, and how they affect the function's behavior as x approaches positive and negative infinity.

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