Question 80
Consider the function
$$
f(x) = (a^{2} + 4a + 2)x^{3} + (a^{3} + 4a^{2} + a + 1)x^{2} + (2a - a^{2})x + a^{2}
$$
where $a$ is a real parameter.
a) Prove that $f(-a) = 0$.
b) Find all values of $a$ such that the equation $f(x) = 0$ has three different positive roots.
$$
f(x) = (a^{2} + 4a + 2)x^{3} + (a^{3} + 4a^{2} + a + 1)x^{2} + (2a - a^{2})x + a^{2}
$$
where $a$ is a real parameter.
a) Prove that $f(-a) = 0$.
b) Find all values of $a$ such that the equation $f(x) = 0$ has three different positive roots.