How can you find the coefficients of a monic quadratic when you know only one non-real root?

We know that non-real roots appear in complex conjugate pairs. Hence when we know one root, we know both of them. Then, as we can factorise a quadratic in it's linear factors, we know our quadratic is a constant times the product of x minus the roots. Lastly, as our quadratic is monic, we must have that this constant is 1. Then find the coefficients by expanding the brackets.