Q. \[ x^2 – 4x + 3 \]

We are asked to work with the expression \(x^2 – 4x + 3\). A common and useful step with a quadratic like this is to factor it. Factoring helps you understand roots (solutions to \(x^2 – 4x + 3 = 0\)) and simplifies many problems.

Step 1: Identify the quadratic form

The expression is in the form

\[
x^2 – 4x + 3
\]

So the coefficients are: \(a = 1\), \(b = -4\), and \(c = 3\).

Step 2: Find two numbers that multiply to \(c\)

We want two numbers whose product is \(c = 3\) and whose sum is \(b = -4\).

The factor pairs of \(3\) are:

\[
1 \cdot 3 = 3
\]

To make the sum \( -4\), we need negative numbers:

\[
(-1) + (-3) = -4
\]

And their product is:

\[
(-1)(-3) = 3
\]

So the two numbers are \(-1\) and \(-3\).

Step 3: Split and factor by grouping

Rewrite the middle term \(-4x\) using \(-1\) and \(-3\):

\[
x^2 – 4x + 3 = x^2 – x – 3x + 3
\]

Now group the terms:

\[
x^2 – x – 3x + 3 = (x^2 – x) + (-3x + 3)
\]

Factor each group

\[
x^2 – x = x(x – 1)
\]

\[
-3x + 3 = -3(x – 1)
\]

So the expression becomes:

\[
x(x – 1) – 3(x – 1)
\]

Now factor out the common binomial \((x – 1)\):

\[
x(x – 1) – 3(x – 1) = (x – 1)(x – 3)
\]

Final Answer

\[
x^2 – 4x + 3 = (x – 1)(x – 3)
\]