algebra precalculus - Showing that roots of a quadratic polynomial are of opposite signs.

Prove that, for all values of $k$, the roots of the quadratic polynomial $x^2 - (2 + k) x - 3$ are real. Show further that the roots are of opposite signs.

For the first part I was able to demonstrate such by using the discriminant of the quadratic, then using the discriminant of the discriminant.

For the second part I was not able to demonstrate such.

Answer

A general quadratic with roots $\alpha$ and $\beta$ can be written
$$
x^2-(\alpha+\beta)x+\alpha\beta=0
$$
The last, constant term is the product of the roots. In your case that equals $-3$. The roots must therefore have opposite signs.