Q. \(x^2+5x+4\)

We want to work with the polynomial \(x^2 + 5x + 4\). A common and useful step is to factor it, because factoring reveals the roots and helps with other algebra tasks.

Step 1: Identify the form to factor

The expression \(x^2 + 5x + 4\) is a quadratic of the form \(ax^2 + bx + c\), where:

\(a = 1\), \(b = 5\), and \(c = 4\).

Step 2: Factor using the “find two numbers” method

We look for two numbers \(m\) and \(n\) such that:

• \(m + n = 5\)
• \(mn = 4\)

Step 3: Choose the correct numbers

List the factor pairs of \(4\):

• \(1 \cdot 4 = 4\)
• \((-1) \cdot (-4) = 4\)

Now check sums:

• \(1 + 4 = 5\)
• \((-1) + (-4) = -5\)

The pair that gives the sum \(5\) is \(1\) and \(4\).

Step 4: Write the factored form

Substitute these values into the factor structure:

\[x^2 + 5x + 4 = (x + 1)(x + 4).\]

Final Answer

The expression factors as:

\[x^2 + 5x + 4 = (x + 1)(x + 4).\]