We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by q 0 ( z ) = 0, q 1 ( z ) = 1, and q n ( z ) = zq n −1 q n −2 + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two methods. One method uses a recursively constructed matrix, where elements are 0, 1, or −1, whose eigenvalues are the roots of q n ( z ). The other method uses a special-purpose homotopy continuation method, where the solution of the differential equation, [EQUATION], in which the initial condition are 0, and the roots of q n −1 and q n −2 , are also the roots of the Fibonacci-Mandelbrot polynomials.