If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount:
$\textbf{(A)}\ {\pm 2ax + a^2}\qquad  \textbf{(B)}\ {2ax \pm a^2}\qquad  \textbf{(C)}\ {\pm a^2 - 3} \qquad  \textbf{(D)}\ {(x + a)^2 - 3}\qquad\\  \textbf{(E)}\ {(x - a)^2 - 3}$
Let us represent the increase or decrease in $x$ by $(x \pm a)$
Thus our original expression becomes
\[(x \pm a)^2 - 3\]
\[x^2 \pm 2ax + a^2 - 3\]
The absolute difference between these two expressions is $\pm 2ax + a^2$.
Therefore, the answer is $\fbox{(A)}$