If $a \neq b, a^3 - b^3 = 19x^3$, and $a-b = x$, which of the following conclusions is correct?
$\textbf{(A)}\ a=3x \qquad  \textbf{(B)}\ a=3x \text{ or } a = -2x \qquad  \textbf{(C)}\ a=-3x \text{ or } a = 2x \qquad \\ \textbf{(D)}\ a=3x \text{ or } a=2x \qquad  \textbf{(E)}\ a=2x$
We can factor $a^3-b^3=19x^3$ into:
\[(a-b)(a^2+ab+b^2)=19x^3.\]
Substituting yields:
\[x(a^2+ab+b^2)=19x^3\]
\[a^2+ab+b^2=19x^2.\]
This is equal to:
\[(a-b)^2+3ab=19x^2\]
\[x^2+3ab=19x^2\]
\[ab=6x^2.\]
Checking with the possible answers, along with $a-b=x$ yields the only answer to be $\boxed{(\textbf{B}): a=3x\text{ or }a=-2x}.$