Sure, I can detail this step by step.
1. Let's first identify what the degree of a polynomial is. The degree of a polynomial is the highest power of x within that polynomial.
2. Examining the function f(x) = -5x - 10 + 3x^4 + 4x^2, we see a few different terms. Our job is to identify which term has the highest power of x.
3. The terms of the given function are -5x, -10, 3x^4, and 4x^2. In these, we see x's powers are 1 (for -5x), 0 (for -10 as it has no x), 4 (for 3x^4), and 2 (for 4x^2).
4. Evidently, the highest power of x is 4. Therefore, the degree of this polynomial is 4.
5. Now, let's arrange the polynomial into the standard form. The standard form of a polynomial is structured so that powers of x are in decreasing order.
6. Sorting the terms of the function in decreasing order of x powers, we start from the highest, which is 4 for the term 3x^4. It's followed by the term with x raised to power 2, which is 4x^2. We then pick the term with power 1, -5x, and finish with the constant term, -10.
7. Consequently, the polynomial is arranged into standard form: f(x) = 3x^4 + 4x^2 - 5x - 10.
To sum it up, the degree of the polynomial is 4, and the polynomial in standard form is f(x) = 3x^4 + 4x^2 - 5x - 10.