Q. \(x^2 – 7x – 18\).

We are factoring the polynomial \(x^2 – 7x – 18\) step by step.

Step 1: Identify the structure

The expression \(x^2 – 7x – 18\) has the form

\[
x^2 + bx + c
\]

So we identify:

• \(b = -7\)
• \(c = -18\)

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

We need two integers \(m\) and \(n\) such that:

• \(m \cdot n = -18\)
• \(m + n = -7\)

Now list factor pairs of \(-18\):

• \( -1 \cdot 18 = -18\) and \(-1 + 18 = 17\)
• \( -2 \cdot 9 = -18\) and \(-2 + 9 = 7\)
• \( -3 \cdot 6 = -18\) and \(-3 + 6 = 3\)
• \( 3 \cdot -6 = -18\) and \(3 + (-6) = -3\)
• \( 2 \cdot -9 = -18\) and \(2 + (-9) = -7\)
• \( 1 \cdot -18 = -18\) and \(1 + (-18) = -17\)

The pair that works is:

• \(m = 2\)
• \(n = -9\)

because \(2 \cdot (-9) = -18\) and \(2 + (-9) = -7\).

Step 3: Rewrite the middle term using the found numbers

Replace \(-7x\) with \(2x – 9x\):

\[
x^2 – 7x – 18 = x^2 + 2x – 9x – 18
\]

Step 4: Factor by grouping

Group the terms into two pairs:

\[
x^2 + 2x – 9x – 18 = \left(x^2 + 2x\right) + \left(-9x – 18\right)
\]

Factor each group:

\[
x^2 + 2x = x(x + 2)
\]
\[
-9x – 18 = -9(x + 2)
\]

So the expression becomes:

\[
x(x + 2) – 9(x + 2)
\]

Step 5: Factor out the common binomial

\(x + 2\) is common in both terms, so factor it out:

\[
x(x + 2) – 9(x + 2) = (x – 9)(x + 2)
\]

Final Answer

\[
x^2 – 7x – 18 = (x – 9)(x + 2)
\]