#### Answer

The polynomial's degree is $6$.

#### Work Step by Step

RECALL:
(1) The degree of a term is equal to the sum of the exponents of its variables.
(2) The degree of a polynomial is equal to the degree of the term with the highest degree.
The terms of the given polynomial have the following degrees:
$-x^4$: degree = 4
$5x^3$: degree = 3
$3x^6$: degree = 6
$-1$: degree = 0
The term with the highest degree is $3x^6$.
The degeree of this term is $6$.
Thus, the polynomial's degree is $6$.