Let's break down and solve the given polynomial expression:

Step 1: Distribute the negative sign and combine like terms

The given expression is:

$(3 + 7x^3 + x^4) - (8 - x + x^4)$

Now distribute the negative sign to each term in the second polynomial:

$3 + 7x^3 + x^4 - 8 + x - x^4$

Step 2: Combine like terms

1. Combine the constant terms: $3 - 8 = -5$

2. Combine the $x$ terms: $+x \quad (\text{there's only one }x\text{-term})$

3. Combine the $x^3$ terms: $7x^3 \quad (\text{there's only one }x^3\text{-term})$

4. Combine the $x^4$ terms: $x^4 - x^4 = 0$

Thus, the resulting polynomial is:

$-5 + x + 7x^3$

Step 3: 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 of $x$ is $x^3$, so the degree is:

$\text{Degree} = 3$

Step 4: Classify by the number of terms

This polynomial has three terms: $-5$, $x$, and $7x^3$. A polynomial with three terms is called a trinomial.

Final Answer:

• Sum/Difference: $-5 + x + 7x^3$
• Degree: 3
• Classification: Trinomial

Would you like more details or further explanation?

Here are 5 related questions for further exploration:

1. How do you handle subtraction of polynomials with negative exponents?
2. What is the degree of a polynomial with several variables (e.g., $x^2y^3$)?
3. Can a polynomial have more than one highest-degree term?
4. How do you classify a polynomial with only one term?
5. What is the role of the leading coefficient in determining the behavior of a polynomial?

Tip: Always double-check that you've combined like terms, especially when subtracting polynomials with the same variable powers.