Let's solve the equations visible in the image step by step:

6. $(x - 4)^2 = 20$

Solution:

1. Take the square root of both sides: $x - 4 = \pm \sqrt{20} = \pm 2\sqrt{5}$
2. Solve for $x$: $x = 4 \pm 2\sqrt{5}$ Thus, the solutions are: $x = 4 + 2\sqrt{5}, \quad x = 4 - 2\sqrt{5}$

7. $\left(\frac{3x + 2}{5}\right)^2 = 144$

Solution:

1. Take the square root of both sides: $\frac{3x + 2}{5} = \pm \sqrt{144} = \pm 12$
2. Multiply both sides by 5: $3x + 2 = \pm 60$
3. Solve for $x$ by considering both cases:
   • For $3x + 2 = 60$: $3x = 58 \implies x = \frac{58}{3}$
   • For $3x + 2 = -60$: $3x = -62 \implies x = \frac{-62}{3}$ Thus, the solutions are: $x = \frac{58}{3}, \quad x = \frac{-62}{3}$

8. $(2x - 6)^2 + 7 = 71$

Solution:

1. Subtract 7 from both sides: $(2x - 6)^2 = 64$
2. Take the square root of both sides: $2x - 6 = \pm 8$
3. Solve for $x$ in both cases:
   • For $2x - 6 = 8$: $2x = 14 \implies x = 7$
   • For $2x - 6 = -8$: $2x = -2 \implies x = -1$ Thus, the solutions are: $x = 7, \quad x = -1$

9. $\left(\frac{5x + 1}{3}\right)^2 - 2 = 1$

Solution:

1. Add 2 to both sides: $\left(\frac{5x + 1}{3}\right)^2 = 3$
2. Take the square root of both sides: $\frac{5x + 1}{3} = \pm \sqrt{3}$
3. Multiply both sides by 3: $5x + 1 = \pm 3\sqrt{3}$
4. Solve for $x$: $5x = -1 \pm 3\sqrt{3}$ $x = \frac{-1 \pm 3\sqrt{3}}{5}$ Thus, the solutions are: $x = \frac{-1 + 3\sqrt{3}}{5}, \quad x = \frac{-1 - 3\sqrt{3}}{5}$

10. $\frac{9(-2x + 4)^2}{10} - 6 = 3$

Solution:

1. Add 6 to both sides: $\frac{9(-2x + 4)^2}{10} = 9$
2. Multiply both sides by 10: $9(-2x + 4)^2 = 90$
3. Divide by 9: $(-2x + 4)^2 = 10$
4. Take the square root of both sides: $-2x + 4 = \pm \sqrt{10}$
5. Solve for $x$: $-2x = -4 \pm \sqrt{10} \implies x = 2 \mp \frac{\sqrt{10}}{2}$ Thus, the solutions are: $x = 2 + \frac{\sqrt{10}}{2}, \quad x = 2 - \frac{\sqrt{10}}{2}$

Do you need further details or explanations?
Here are 5 related questions you might be interested in:

1. How do you solve quadratic equations using the quadratic formula?
2. What are the rules for dealing with square roots when solving equations?
3. How can we check the validity of solutions for quadratic equations?
4. What are the steps for solving equations with fractions or radicals?
5. What is the method of completing the square for solving quadratic equations?

Tip: Always check if the solutions make sense by substituting them back into the original equation!