To factor the quadratic polynomial $x^2 - 4x + 3$, we are looking for two numbers that:

1. Multiply to give the constant term, which is 3.
2. Add up to give the coefficient of the linear term, which is -4.

Let's find two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3, because:

• $-1 \times -3 = 3$
• $-1 + -3 = -4$

Thus, we can rewrite the quadratic as:

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

So, the completely factored form of $x^2 - 4x + 3$ is:

$(x - 1)(x - 3)$

Would you like more details or clarification on any step? Here are some related questions:

1. How do you factor quadratics with no linear term?
2. What happens if the leading coefficient is not 1 in a quadratic?
3. How do you apply the AC method for factoring quadratics?
4. How can you verify your factorization is correct?
5. How can you factor polynomials with more than two terms?

Tip: Always check if the leading coefficient (the number in front of $x^2$) is 1 when factoring quadratics. It makes the process simpler!