Answer:
Let's assume that k represents any real number, let's say k = 3
f(x) = 3+2 x+x^3
Let's take two numbers : x = -30 and x = 12
Then,
f(-30) = -27057 < 0
f(12) = 1755 > 0
By the Intermediate Value Theorem, there exists a number "c" between -30 and 12 such that f(c) = 0. So, the equation has a real root.
Now, let's suppose that there exists two roots w and q. Then, f(w) = f(q) = 0. However, using the Rolle's Theorem, f'(s) = 0 for some s in the set of numbers (w,q), but f'(x) = 2+3 x^2 > 0 for all x. So, it is impossible to have f'(s) = 0 for some c; in other words, there will be no maximum or minimum critical points, which means that there will be exactly one real root
Therefore, we conclude that for any function of family f(x) = x^3 + 2x + k, where k is any real number, there exists exactly one real root