Suppose that $f$ is a polynomial such that
\[
(x-1) \cdot f(x)=3 x^{4}+x^{3}-25 x^{2}+38 x-17
\]
What is the degree of $f ? m$ is a real number and $2 x^{2}+m x+8$ has two distinct real roots, then what are the possible values of $m$ ? Express your answer in interval notation.

Step 1 ：First, we find the degree of f(x) using the same method as in SolutionA. Since the product of f and a polynomial with degree 1 equals a polynomial with degree 4, we know that f is a polynomial with degree \(4-1=\boxed{3}\).

Step 2 ：Now, we need to find the possible values of m for which the quadratic equation \(2x^2+mx+8\) has two distinct real roots. To do this, we use the discriminant, which is given by \(D = b^2 - 4ac\), where a=2, b=m, and c=8.

Step 3 ：Calculate the discriminant: \(D = m^2 - 4(2)(8) = m^2 - 64\).

Step 4 ：For the quadratic equation to have two distinct real roots, the discriminant must be greater than 0. So, we have \(m^2 - 64 > 0\).

Step 5 ：To solve this inequality, we can factor the left side: \((m-8)(m+8) > 0\).

Step 6 ：Using the sign analysis method, we find that the inequality holds when \(m<-8\) or \(m>8\).

Step 7 ：So, the possible values of m are expressed in interval notation as \(\boxed{(-\infty, -8) \cup (8, \infty)}\).