Solution:

The answer is (E). We can proceed by elimination. (A) is false because if $x>0$ then at each turn Alberto and Barbara multiply and add positive numbers to those already on the blackboard. Since the first two numbers are $x$ and $1$, then there is no negative number on the blackboard. (B) and (C) are false because $x$ can be negative. (D) is false because $x$ can be positive.

To show that $(\mathbf{E})$ is indeed true, we observe that the first numbers written by Barbara are $1, 1+x+x^{2}, 1+x+x^{2}+x^{3}+x^{4} = 1+x+x^{2}(1+x+x^{2})$, and that in general Barbara will write expressions of the form $1+x+\ldots+x^{2n}$. A number of this type is necessarily positive. Indeed, if $x=1$ this is obvious, and otherwise the sum can be computed using the properties of geometric progressions: we have $1+x+\ldots+x^{2n} = \frac{x^{2n+1}-1}{x-1}$, which is positive because the numerator and denominator have the same sign (positive if $x>1$, negative if $x<1$).