Problem 269

The answer is (B). First, observe that the polynomial in the statement can be written as $(1+2 x+2 x^{2}+x^{3})+(x^{3}+2 x^{4}+2 x^{5}+x^{6}) = (1+2 x+2 x^{2}+x^{3}) + x^{3}(1+2 x+2 x^{2}+x^{3}) = (1+x^{3})(1+2 x+2 x^{2}+x^{3})$.

Recognizing further that we can write $1+2 x+2 x^{2}+x^{3} = (1+x+x^{2}) + (x+x^{2}+x^{3}) = (1+x)(1+x+x^{2})$, the problem reduces to solving the equation $(x^{3}+1)(x+1)(1+x+x^{2})=0$.

Since a product is zero when one of the factors is zero, the solutions are those of the equation $x^{3}=-1$, namely $x=-1$, those of $x+1=0$, namely again $-1$, and those of $1+x+x^{2}=0$, which has no solutions, since $1+x+x^{2} = \left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}$ is always positive. We conclude therefore that the answer is 1: the only real solution of the proposed equation is $x=-1$.