If $p(x) = ax^3 + bx^2 + cx + d $ and if $a$ and $d$ happen to be of same sign, then you can conclude that there must be at least one negative root. Probably this is what your friend would have done as well.
EDIT
The question was changed from having a negative root to there is at least one real root.
This is a consequence of the fact that $p(x)$ is continuous and the intermediate value theorem for continuous functions.
The important observation is that $$\lim_{x \rightarrow \infty} p(x) = \text{sign(a)} \cdot \infty$$
and
$$\lim_{x \rightarrow -\infty} p(x) = -\text{sign(a)} \cdot \infty$$
For instance, if $a > 0$, then $\displaystyle \lim_{x \rightarrow \infty} p(x) = \infty$
and $\displaystyle \lim_{x \rightarrow -\infty} p(x) = -\infty$.
Since the function is continuous, by intermediate value theorem, it must hit all values between $- \infty$ and $\infty$ for some $x$. Hence, in particular, it must hit $0$. Intutively, the function is positive for some large $x$ and negative for some large $x$ and since the function has no jumps, it must hit $0$ somewhere on the real axis before it changes sign.