4A. Let the equation $a x^{2}+b x+c=0$, where $a \neq 0$, have no real solutions. Determine the sign of $c$, given that $a+b+c<0$.
Solution. According to the condition of the problem, the quadratic function

$$
f(x)=a x^{2}+b x+c
$$

has no real zeros, so $f(x)0$ for every $x \in \mathbb{R}$. But $f(1)=a+b+c<0$, so it must be that $f(0)=c<0$.

ZB. Simplify the expression

$$
A=\left(1+\frac{1+i}{2}\right) \cdot\left(1+\left(\frac{1+i}{2}\right)^{2}\right) \cdot\left(1+\left(\frac{1+i}{2}\right)^{4}\right) \cdot\left(1+\left(\frac{1+i}{2}\right)^{8}\right)
$$

$i$ is the imaginary unit.