Let's solve the given task step by step:

We are tasked to determine:

1. Nullstellen (zeros) of the functions $f(x) = x^2 + 5x + 4$ and $g(x) = 0.5x - 1$.
2. Schnittpunkte (intersection points) of $f$ and $g$.

1. Nullstellen (zeros) of $f(x) = x^2 + 5x + 4$

To find the zeros of a quadratic function, we solve: $f(x) = 0 \quad \Rightarrow \quad x^2 + 5x + 4 = 0.$

This can be factored as: $x^2 + 5x + 4 = (x + 4)(x + 1) = 0.$

From the factored form, the solutions are: $x_1 = -4 \quad \text{and} \quad x_2 = -1.$

2. Nullstelle of $g(x) = 0.5x - 1$

To find the zero of $g(x)$, set $g(x) = 0$: $0.5x - 1 = 0.$

Solve for $x$: $0.5x = 1 \quad \Rightarrow \quad x = 2.$

Thus, the zero of $g(x)$ is: $x = 2.$

3. Schnittpunkte (intersections) of $f$ and $g$

To find the intersection points, set $f(x) = g(x)$, i.e.: $x^2 + 5x + 4 = 0.5x - 1.$

Reorganize into standard quadratic form: $x^2 + 5x + 4 - 0.5x + 1 = 0 \quad \Rightarrow \quad x^2 + 4.5x + 5 = 0.$

Now solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$ Here:

• $a = 1$,
• $b = 4.5$,
• $c = 5$.

Substitute these values: $x = \frac{-4.5 \pm \sqrt{(4.5)^2 - 4(1)(5)}}{2(1)}.$

Simplify step by step: $x = \frac{-4.5 \pm \sqrt{20.25 - 20}}{2}.$ $x = \frac{-4.5 \pm \sqrt{0.25}}{2}.$ $x = \frac{-4.5 \pm 0.5}{2}.$

Now calculate the two solutions:

1. $x_1 = \frac{-4.5 + 0.5}{2} = \frac{-4}{2} = -2$,
2. $x_2 = \frac{-4.5 - 0.5}{2} = \frac{-5}{2} = -2.5$.

Thus, the $x$-coordinates of the intersection points are $x_1 = -2$ and $x_2 = -2.5$.

To find the corresponding $y$-values, substitute these $x$-values into $g(x) = 0.5x - 1$:

1. For $x = -2$: $y = 0.5(-2) - 1 = -1 - 1 = -2.$
2. For $x = -2.5$: $y = 0.5(-2.5) - 1 = -1.25 - 1 = -2.25.$

Thus, the intersection points are: $S_1(-2, -2) \quad \text{and} \quad S_2(-2.5, -2.25).$

Final Answers:

1. Nullstellen:

   • $f(x)$: $x_1 = -4, x_2 = -1$,
   • $g(x)$: $x = 2$.

2. Schnittpunkte:

   • $S_1(-2, -2)$,
   • $S_2(-2.5, -2.25)$.

Any questions or further explanations needed?

Here are related questions for you to explore further:

1. How do you solve quadratic equations using the quadratic formula?
2. What does the intersection of two functions represent graphically?
3. How can you verify solutions for intersection points?
4. How do you determine the vertex of a quadratic function?
5. How do linear and quadratic functions behave differently graphically?

Tip: Always check your solutions by substituting back into the original equations.