is x+10 a polynomial with integer coefficients?

Yes.

It is a polynomial, and the coefficients of the terms are integers, therefore it is a polynomial with integer coefficients.

A monomial has 1 term
A binomial has 2 terms
A trinomial has 3 terms
A polynomial has many terms (1 or more) and includes all the cases above.

A term has a coefficient (a number in front, usually an integer) then a variable and then some integer power.

In your case, the first term is:
x --> coefficient = 1, variable = x, power = 1.
10 --> coefficient = 10, variable = x, power = 0.

So yes, you have everything you need to consider that a polynomial with integer coefficients.