Let's work through the table based on the quadratic equations provided. We will apply the formulas for the sum and product of the roots of a quadratic equation:

For a quadratic equation in the form $ax^2 + bx + c = 0$:

• Sum of the roots: $x_1 + x_2 = -\frac{b}{a}$
• Product of the roots: $x_1 x_2 = \frac{c}{a}$

Let's go row by row:

1. $x^2 + 5x + 4 = 0$

• Here, $a = 1$, $b = 5$, and $c = 4$.
  • Sum of the roots: $x_1 + x_2 = -\frac{b}{a} = -\frac{5}{1} = -5$
  • Product of the roots: $x_1 x_2 = \frac{c}{a} = \frac{4}{1} = 4$

2. $x^2 - 10x - 24 = 0$

• Here, $a = 1$, $b = -10$, and $c = -24$.
  • Sum of the roots: $x_1 + x_2 = -\frac{b}{a} = -\frac{-10}{1} = 10$
  • Product of the roots: $x_1 x_2 = \frac{c}{a} = \frac{-24}{1} = -24$

3. $6x^2 + 12x - 18 = 0$

• Here, $a = 6$, $b = 12$, and $c = -18$.
  • Sum of the roots: $x_1 + x_2 = -\frac{b}{a} = -\frac{12}{6} = -2$
  • Product of the roots: $x_1 x_2 = \frac{c}{a} = \frac{-18}{6} = -3$

4. $9x^2 - 6x = 0$

• Here, $a = 9$, $b = -6$, and $c = 0$.
  • Sum of the roots: $x_1 + x_2 = -\frac{b}{a} = -\frac{-6}{9} = \frac{6}{9} = \frac{2}{3}$
  • Product of the roots: $x_1 x_2 = \frac{c}{a} = \frac{0}{9} = 0$

Let me know if you'd like any further clarifications!

Here are 5 follow-up questions you could explore based on this:

1. How do we factor the given quadratic equations to verify the roots?
2. Can we graph each of these quadratic equations to visualize their roots?
3. How do complex roots affect the sum and product of roots in these types of problems?
4. What is the relationship between the discriminant and the product of the roots?
5. Can the sum or product of the roots ever be greater than the coefficients in certain quadratic equations?

Tip: Always check the sign of the coefficients carefully when calculating the sum and product of roots to avoid mistakes.