The monic quintic polynomials with one real root form a manifold I am required to determine whether or not the set of monic quintics with exactly one real root is a manifold.
I do not know how to do this. If I think of the set of monic quintic polynomials in general, I think of a natural mapping from $p(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ to $(a, b, c, d, e) \in R^5$. Each monic quintic is uniquely determined by these coefficients, and so this is an injective mapping.
I also know that if $p(x)$ has exactly one real root that its other four roots consist of a pair of complex conjugates $z_1 , z_2 , \overline z_1 , \overline z_2$. It factors into a linear term and two quadratic terms.
I think it would be pertinent to first show that the set of monic quintics is itself a manifold, but I do not even see how to do this. If the only chart is the map above from each $p(x)$ to its coefficients, do we trivially have a manifold?
 A: Implicit here is that you are supposed to topologize the set of monic quintics as $\mathbb{R}^5$ via the coefficient map as you describe, so the entire set of monic quintics is trivially a manifold, namely $\mathbb{R}^5$ itself.
From here you can show that the space of monic quintics with exactly one real root is open in the space of all monic quintics, so in particular it's an open submanifold. (Jyrki's comment shows that it's not closed, but that doesn't matter.) Here is the intuitive idea: if such a polynomial $f(x)$ has exactly one real root then there is some $\varepsilon > 0$ such that its other four complex roots are at least a distance of $\varepsilon$ from the real line. Then if you wiggle $f(x)$ a small amount its roots wiggle a small amount, and you can arrange to wiggle $f(x)$ small enough so that each of its complex roots continues to not touch the real line. This shows that there's some open neighborhood around $f(x)$ also consisting of polynomials with four non-real roots (meaning the remaining root must be real).
This intuitive argument can be made precise by showing that the roots of a polynomial depend continuously on its coefficients in a suitable sense. It's a little delicate to make "suitable sense" precise because the roots don't come in a prescribed order and can have multiplicities, but with multiplicities taken into account it's still true (the roots can collide and separate but they still do so continuously).
