quadratic

Let's consider the quadratic equation $x^2 + (b + 4c)x + c^2 = 0$.

For a quadratic equation $ax^2 + bx + c = 0$ to have exactly one solution, its discriminant ($b^2 - 4ac$) must be equal to $0$.

In this case, the discriminant is $(b + 4c)^2 - 4 \cdot 1 \cdot c^2 = b^2 + 8bc + 16c^2 - 4c^2$.

Simplify the discriminant:

$$b^2 + 8bc + 12c^2.$$

For the quadratic equation to have exactly one solution, the discriminant must be equal to $0$:

$$b^2 + 8bc + 12c^2 = 0.$$

Now, we want to find the non-zero value of $c$ for which there is exactly one positive value of $b$ that satisfies this equation.

Notice that the quadratic equation above is a quadratic in $b$, and we want it to have a single solution. This means the discriminant of this quadratic in $b$ must be equal to $0$:

$$8^2 - 4 \cdot 12c^2 = 0.$$

Solve for $c$:

$$64 - 48c^2 = 0 \Rightarrow 48c^2 = 64 \Rightarrow c^2 = \frac{64}{48} = \frac{4}{3}.$$

Take the positive square root:

$$c = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}.$$

So, the non-zero value of $c$ that satisfies the given conditions is $\boxed{\frac{2\sqrt{3}}{3}}$.