Prove that any polynomial with real coefficients can be factored into polynomials of degrees one and two, with real coefficients.

Prove that any polynomial with real coefficients can be factored into polynomials of degrees one and two, with real coefficients. (Please make sure to indicate the following ideas as formally as possible: induction on the power (or degree) of the polynomial, the Fundamental Theorem of Algebra, the properties of the conjugates, and the long division of polynomials.)