From the image, it looks like the task is to solve quadratic equations by factoring. Let’s break down the steps based on the problems shown.

Problem 1:

$-2x^2 + 12x - 18 = 0$

Solution (Factoring method):

1. Simplify by factoring out common terms: Factor out $-2$: $-2(x^2 - 6x + 9) = 0$

2. Factor the quadratic: $x^2 - 6x + 9$ is a perfect square trinomial: $x^2 - 6x + 9 = (x - 3)^2$

   Substituting: $-2(x - 3)^2 = 0$

3. Solve for $x$: Divide through by $-2$: $(x - 3)^2 = 0$

   Taking the square root: $x - 3 = 0$

   Solve: $x = 3$

   Solution: $x = 3$.

Problem 2:

$x^2 - 4x + 3 = 0$

Solution (Factoring method):

1. Factor the quadratic: Find two numbers that multiply to $3$ and add to $-4$: These are $-3$ and $-1$. $x^2 - 4x + 3 = (x - 3)(x - 1)$

2. Solve for $x$: Set each factor equal to 0: $x - 3 = 0 \quad \text{or} \quad x - 1 = 0$

   Solve: $x = 3 \quad \text{or} \quad x = 1$

   Solution: $x = 3, x = 1$.

Would you like further explanation of these steps or assistance with similar problems?

5 Follow-up Questions:

1. How do you recognize a perfect square trinomial when factoring?
2. Can all quadratic equations be solved by factoring? Why or why not?
3. What other methods exist for solving quadratic equations besides factoring?
4. How do you check if your solutions are correct after solving a quadratic equation?
5. Can the quadratic formula be used as a general method even when factoring works?

1 Tip:

When factoring, always check for common factors across all terms before attempting more complex methods.