Let's work through the questions based on the image.

Problem 1:

Find the value(s) of $k$ so that the line through $\left(k, -2\right)$ and $\left(-4, 3k\right)$ has a slope of $m = k + 1$.

To find the slope between two points, we use the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the coordinates $\left(k, -2\right)$ and $\left(-4, 3k\right)$: $\frac{3k - (-2)}{-4 - k} = k + 1$ Simplifying the slope expression: $\frac{3k + 2}{-4 - k} = k + 1$ Now solve for $k$ by cross-multiplying: $3k + 2 = (k + 1)(-4 - k)$ Expand the right side: $3k + 2 = -4(k + 1) - k(k + 1)$ $3k + 2 = -4k - 4 - k^2 - k$ Combine like terms: $3k + 2 = -k^2 - 5k - 4$ Bring all terms to one side: $k^2 + 8k + 6 = 0$ Solve this quadratic equation using the quadratic formula: $k = \frac{-8 \pm \sqrt{8^2 - 4(1)(6)}}{2(1)}$ $k = \frac{-8 \pm \sqrt{64 - 24}}{2}$ $k = \frac{-8 \pm \sqrt{40}}{2}$ $k = \frac{-8 \pm 2\sqrt{10}}{2}$ $k = -4 \pm \sqrt{10}$ Thus, the two solutions for $k$ are: $k = -4 + \sqrt{10} \quad \text{or} \quad k = -4 - \sqrt{10}$

Would you like more detailed explanations for any of these steps, or shall I move on to the next problem?

Related questions:

1. How do you solve quadratic equations using the quadratic formula?
2. What is the general formula for the slope of a line between two points?
3. How can you determine if two lines are parallel or perpendicular based on their slopes?
4. How do you simplify expressions involving radicals?
5. What are the properties of a line's slope in terms of different coordinate systems?

Tip:

Always double-check the signs and arithmetic when solving equations involving variables on both sides to avoid common mistakes.