The problem asks for solving quadratic equations in two parts:

Problem 4:

You are tasked with factoring and solving the quadratic equation:

$2x^2 - 5x - 3 = 0$

Steps to solve:

1. Multiply $2$ (the coefficient of $x^2$) and $-3$ (the constant term), which equals $-6$.
2. Find two numbers that multiply to $-6$ and add to $-5$ (the middle term coefficient). The numbers are $-6$ and $1$.
3. Rewrite the middle term: $2x^2 - 6x + x - 3 = 0$.
4. Group terms: $(2x^2 - 6x) + (x - 3) = 0$.
5. Factor out common terms: $2x(x - 3) + 1(x - 3) = 0$.
6. Factor the equation: $(2x + 1)(x - 3) = 0$.
7. Solve for $x$: $x = -\frac{1}{2}$ or $x = 3$.

The factored equation is $(2x + 1)(x - 3) = 0$ and the solutions are $x = -\frac{1}{2}$ and $x = 3$.

Problem 5:

You are asked to solve the equation using the Square Root Property or Completing the Square. The given equation is:

$x^2 - 12x + 36 = 64$

Steps to solve:

1. Rewrite the equation: $(x - 6)^2 = 64$ (since $x^2 - 12x + 36$ is a perfect square trinomial).
2. Take the square root of both sides: $x - 6 = \pm 8$.
3. Solve for $x$:
   • $x - 6 = 8 \Rightarrow x = 14$,
   • $x - 6 = -8 \Rightarrow x = -2$.

The solutions are $x = 14$ and $x = -2$.

Do you need detailed explanations for these steps or further clarification? Here are related questions to deepen your understanding:

1. Why is factoring a useful method for solving quadratic equations?
2. How do you determine when to use the Square Root Property versus Completing the Square?
3. What does it mean geometrically when solving $(x - 6)^2 = 64$?
4. Can every quadratic equation be solved by factoring? Why or why not?
5. What are some common mistakes to avoid when factoring or completing the square?

Tip: Always check your solutions by substituting them back into the original equation to ensure they satisfy it.