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To analyze the given polynomial:

The function is:
\[
f(x) = 2x^5 + 10x^4 - 6x^3 - 90x^2 - 108x
\]

**Step 1: Identify the degree of the polynomial.**
The degree of a polynomial is the highest power of \( x \) in the expression. In this case, the highest power is \( x^5 \).

**Degree:** \( 5 \)

**Step 2: Identify the leading coefficient.**
The leading coefficient is the coefficient of the term with the highest power of \( x \). Here, the term with \( x^5 \) has a coefficient of \( 2 \).

**Leading coefficient:** \( 2 \)

### Final Answer
The degree of the polynomial is **5**, and the leading coefficient is **2**.

Would you like more details or have any questions?

### Here are 5 related questions:
1. What is the behavior of this polynomial as \( x \to \infty \) and \( x \to -\infty \)?
2. How many roots can a 5th-degree polynomial have?
3. Can you identify any potential rational roots using the Rational Root Theorem?
4. What are the turning points of a polynomial of degree 5?
5. How would you factor a polynomial like this one?

### Tip
Always start analyzing a polynomial by identifying the highest degree term and its coefficient, as these define many key properties of the function.

The degree of the polynomial is ___, and the leading coefficient is ___.

Learn how to identify the degree and leading coefficient of the polynomial function f(x) = 2x^5 + 10x^4 - 6x^3 - 90x^2 - 108x. This problem covers key concepts in polynomial functions, including determining the highest power and its coefficient. Suitable for high school students in Grades 10-12.

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