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Dividing by the greatest common factor (GCF) is a strategic simplification, not a required algebraic rule. Here’s why it helps in this problem:

1. Keeps the equation equivalent
   When you divide every term (including the constant term on the right‑hand side) by the same non‑zero number, you are performing the same operation on both sides of the equation. The set of solutions does not change.

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

2. Reduces the size of the coefficients
   Smaller numbers are easier to work with when you look for factor pairs.

   • In the original quadratic the constant term is 6, so you would need two numbers whose product is $2 \times 6 = 12$ (because the leading coefficient is 2).
   • After dividing by 2 the constant term becomes 3, and the leading coefficient becomes 1, so you only need two numbers whose product is 3. That’s much simpler.

3. Makes the factoring step more transparent
   With the simplified form $x^{2}-4x+3$ you can directly search for two numbers that multiply to +3 and add to ‑4. Those numbers are $-1$ and $-3$, giving the factorisation $(x-1)(x-3)$.
   If you tried to factor the original expression without first dividing, you would have to find numbers $p$ and $q$ such that

   $$p\;q = (2)(6)=12 \quad\text{and}\quad p+q = -8$$

   The pair $-2$ and $-6$ works, but you must then rewrite the middle term and factor by grouping, which adds extra steps and a higher chance of arithmetic slip‑ups.

4. Avoids the “divide only part of the equation” mistake
   A common error is to divide just the left‑hand side or only the constant term, which changes the equality and leads to wrong solutions. By explicitly dividing every term (including the zero on the right), you guarantee the equation remains balanced.

Bottom line

Dividing by the GCF first is a convenience that:

• preserves the solution set,
• simplifies the numbers you work with,
• reduces the chance of algebraic errors.

You could solve the quadratic without that step, but the extra arithmetic makes mistakes more likely—especially when you’re still mastering factoring.

Do you want a quick visual recap of the factoring process, or another example where we don’t divide first so you can see the difference?