A quadratic polynomial $a{x}^2+bx+c$ has two real zeroes of opposite signs (one positive, one negative). Which statement must be true?

Zeroes of opposite signs make their product negative, and the product is $\frac{c}{a}$. So $\frac{c}{a}<0$, i.e. a and $c$ must have opposite signs $(ac<0).b=0$ is the stronger condition for zeroes that are equal in magnitude as well as opposite in sign; $b^2=4ac$ is the condition for equal zeroes; and $b$ only controls the sum $-\frac{b}{a}$, which here may be positive, negative or zero.