The equation k = |f(x)| may have 2, 4, 6, 7 and 8 roots. Some of these numbers of roots are obtained when k is not an integer, though.
Let -m be the minimum of $\displaystyle (x^2 - 4)(x^2 - 25)$, i.e., $\displaystyle -m/\lambda$ is the minimum of f(x). According to my calculations, m > f(0) (0 is one of the critical points of f(x)). Therefore, when $\displaystyle k > m/\lambda$, the equation has two roots. From the inequality $\displaystyle 2 > m/\lambda$ you can find the range of $\displaystyle \lambda$ for k = 2. Then find whether $\displaystyle 4 = m/\lambda$ gives an integer $\displaystyle \lambda$. When $\displaystyle f(0) = 100/\lambda < k < m/\lambda$, the equation has 6 roots. Solving $\displaystyle 100/\lambda < 6 < m/\lambda$ gives you the values of $\displaystyle \lambda$ for k = 6, and so on.