How do you write a polynomial in standard form, then classify it by degree and...

Explanation:
Let's combine like terms first (i.e. Anything with ${\displaystyle x^3}$ can be combined, anything with ${\displaystyle x^2}$ can be combined, etc.):
${\displaystyle 12x^3+5-5x^2-6x^2+3x^3+2-x}$
(Resort the numbers into standard form, to make it easier to simplify)
${\displaystyle 12x^3+3x^3-5x^2-6x^2-x+5+2}$
(Combine like terms)
${\displaystyle 15x^3-11x^2-x+7}$
We now have our simplified polynomial. This can be classified as a 3rd degree polynomial
We classify by number of terms by finding how many individual terms there are.
${\displaystyle 15x^3,11x^2,x}$, amd 7 are all four of our terms, so we have a polynomial (Any polynomial with four or more terms is just called a polynomial.)
We classify by degree by finding the highest exponent on any of the terms. The exponent 3 is the highest term, so this polynomial is a 3rd degree polynomial.