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The problem asks if the following functions are second-degree trinomials (also known as quadratic trinomials). Let's analyze both functions one by one.

### Function a)
\[ f(x) = (x - 1)^2 - x^2 \]

Let's expand and simplify this function:

1. Expand \((x - 1)^2\):
   \[
   (x - 1)^2 = x^2 - 2x + 1
   \]

2. Substitute into the expression for \(f(x)\):
   \[
   f(x) = (x^2 - 2x + 1) - x^2
   \]

3. Simplify:
   \[
   f(x) = x^2 - 2x + 1 - x^2 = -2x + 1
   \]

**Conclusion**: The function \(f(x) = -2x + 1\) is a first-degree polynomial (linear), not a second-degree trinomial.

### Function b)
\[ g(x) = x^4 - \frac{1}{x} - x(x^2 + x) \]

Let's simplify this function:

1. Distribute the \(x\) in the term \(-x(x^2 + x)\):
   \[
   g(x) = x^4 - \frac{1}{x} - (x^3 + x^2)
   \]

2. Rearrange the terms:
   \[
   g(x) = x^4 - x^3 - x^2 - \frac{1}{x}
   \]

**Conclusion**: The function \(g(x) = x^4 - x^3 - x^2 - \frac{1}{x}\) is a polynomial of degree 4 with a fractional term. It is not a second-degree trinomial.

### Final Answer:
- **Function \(f(x)\)** is not a second-degree trinomial; it is a first-degree polynomial.
- **Function \(g(x)\)** is not a second-degree trinomial; it is a fourth-degree polynomial with an additional fractional term.

Would you like further details on any of these steps or explanations?

### Related Questions:
1. What is a trinomial, and how does it differ from a general polynomial?
2. How can you determine the degree of a polynomial?
3. What are the key characteristics of a second-degree polynomial?
4. How does simplifying expressions help in identifying the degree of a polynomial?
5. Can you transform a higher-degree polynomial into a lower degree by factoring?

### Tip:
Always simplify and rearrange polynomial expressions to identify the highest power of the variable easily. This helps in determining the degree of the polynomial.

Les fonctions suivantes sont-elles des fonctions trinômes du second degré ? Justifier.
a) f(x) = (x - 1)^2 - x^2
b) g(x) = x^4 - 1/x - x(x^2 + x)

Explore whether the given functions are second-degree trinomials with a step-by-step breakdown. This problem involves expanding and simplifying polynomials to determine their degree, suitable for students in Grades 10-12.

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