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Step‑by‑step solution

The quadratic equation is

1. Factor out the greatest common factor (GCF).
   All terms are divisible by 2, so divide the whole equation by 2 (remember to divide every term):

   $$\frac{2x^{2}}{2}-\frac{8x}{2}+\frac{6}{2}=0\quad\Longrightarrow\quad x^{2}-4x+3=0$$

2. Factor the trinomial $x^{2}-4x+3$.
   Look for two numbers that multiply to +3 (the constant term) and add to −4 (the coefficient of $x$).

   The numbers $-1$ and $-3$ satisfy this because
   $-1\times -3 = 3$ and $-1 + (-3) = -4$.

   Hence

   $$x^{2}-4x+3=(x-1)(x-3)$$

3. Apply the zero‑product property.
   If a product equals zero, at least one factor must be zero:

   $$(x-1)(x-3)=0$$

   So

   $$x-1=0 \quad\text{or}\quad x-3=0$$

4. Solve each simple equation.

   $$x=1 \qquad\text{or}\qquad x=3$$

Final Answer

Why dividing by the GCF first matters

Dividing the entire equation by the GCF (here, 2) simplifies the coefficients, making the factoring step easier and less error‑prone. If you only divide part of the equation, the equality no longer holds, leading to incorrect solutions.

Would you like me to clarify any of these steps further?