1960 AHSME Problem 22

The equality (x+m)^2-(x+n)^2=(m-n)^2, where m and n are unequal non-zero constants, is satisfied by x=am+bn, where:

\begin{array}{l} \textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad \\ \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad \\ \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value} \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.