1959 AHSME Problem 48

Given the polynomial a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n, where n is a positive integer or zero, and a_0 is a positive integer. The remaining a's are integers or zero. Set h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|. [See example 25 for the meaning of |x|.] The number of polynomials with h=3 is: \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9

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