Let's solve these problems step by step:

1. Solve $x^2 - 9x + 20 = 0$ using factorization.

The equation is $x^2 - 9x + 20 = 0$.

• Find two numbers whose product is 20 (constant term) and sum is -9 (coefficient of $x$).
  • The numbers are -5 and -4.
• Rewrite the middle term: $x^2 - 5x - 4x + 20 = 0$
• Group terms and factorize: $x(x - 5) - 4(x - 5) = 0$ $(x - 5)(x - 4) = 0$
• Solve for $x$: $x - 5 = 0 \quad \text{or} \quad x - 4 = 0$ $x = 5 \quad \text{or} \quad x = 4$

Solution: $x = 5, x = 4$

2. Solve $2x^2 + 12x + 10 = 0$ using the complete square method.

The equation is $2x^2 + 12x + 10 = 0$.

1. Divide through by 2 to simplify: $x^2 + 6x + 5 = 0$
2. Rewrite $x^2 + 6x + 5 = 0$ by completing the square:
   • Take half of 6 (coefficient of $x$) and square it: $(6/2)^2 = 9$.
   • Add and subtract 9: $x^2 + 6x + 9 - 9 + 5 = 0$
   • Simplify: $(x + 3)^2 - 4 = 0$
3. Solve for $x$: $(x + 3)^2 = 4$ $x + 3 = \pm 2$ $x = -3 + 2 = -1 \quad \text{or} \quad x = -3 - 2 = -5$

Solution: $x = -1, x = -5$

3a. Solve $6x^2 - 11x - 7 = 0$ using the quadratic formula.

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 6$, $b = -11$, $c = -7$.

1. Calculate the discriminant: $\Delta = b^2 - 4ac = (-11)^2 - 4(6)(-7) = 121 + 168 = 289$
2. Solve for $x$: $x = \frac{-(-11) \pm \sqrt{289}}{2(6)} = \frac{11 \pm 17}{12}$ $x = \frac{11 + 17}{12} = \frac{28}{12} = \frac{7}{3}$ $x = \frac{11 - 17}{12} = \frac{-6}{12} = -\frac{1}{2}$

Solution: $x = \frac{7}{3}, x = -\frac{1}{2}$

3b. Solve $x^2 + 3x - 18 = 0$ using the quadratic formula.

The equation is $x^2 + 3x - 18 = 0$.

1. Calculate the discriminant: $\Delta = b^2 - 4ac = (3)^2 - 4(1)(-18) = 9 + 72 = 81$
2. Solve for $x$: $x = \frac{-3 \pm \sqrt{81}}{2(1)} = \frac{-3 \pm 9}{2}$ $x = \frac{-3 + 9}{2} = \frac{6}{2} = 3$ $x = \frac{-3 - 9}{2} = \frac{-12}{2} = -6$

Solution: $x = 3, x = -6$

Do you need further clarification on any step or method? Here are 5 questions to deepen understanding:

1. What are other methods for solving quadratic equations besides factorization, completing the square, and quadratic formula?
2. How can you verify the roots of a quadratic equation?
3. Can every quadratic equation be solved by factorization? Why or why not?
4. How does the discriminant ($\Delta$) affect the nature of roots?
5. Why is completing the square a useful method for deriving the quadratic formula?

Tip: Always check your solutions by substituting the values of $x$ back into the original equation!