Q. \(x^2+2x-24\)

We want to simplify and solve the expression

\[
x^2 + 2x – 24
\]

Step 1: Look for a way to factor the quadratic

To factor a quadratic of the form

\[
x^2 + 2x – 24
\]

we look for two numbers \(a\) and \(b\) such that:

• \(a \cdot b = -24\)
• \(a + b = 2\)

Step 2: Find numbers that multiply to \(-24\) and add to \(2\)

List factor pairs of \(-24\):

• \(-24\) and \(1\) (sum \(-23\))
• \(-12\) and \(2\) (sum \(-10\))
• \(-8\) and \(3\) (sum \(-5\))
• \(-6\) and \(4\) (sum \(-2\))
• \(-4\) and \(6\) (sum \(2\))
• \(-3\) and \(8\) (sum \(5\))

The pair \(-4\) and \(6\) works because

\[
(-4)(6) = -24
\]

and

\[
-4 + 6 = 2
\]

Step 3: Rewrite the quadratic using those factors

Now factor by splitting the middle term \(2x\) into \(6x – 4x\):

\[
x^2 + 2x – 24 = x^2 + 6x – 4x – 24
\]

Step 4: Factor by grouping

Group terms with common factors:

\[
x^2 + 6x – 4x – 24 = (x^2 + 6x) + (-4x – 24)
\]

Factor each group:

• \[
  x^2 + 6x = x(x + 6)
  \]
• \[
  -4x – 24 = -4(x + 6)
  \]

So the expression becomes:

\[
x(x + 6) – 4(x + 6)
\]

Step 5: Factor out the common binomial \((x + 6)\)

\[
x(x + 6) – 4(x + 6) = (x – 4)(x + 6)
\]

Final Answer (factored form)

\[
x^2 + 2x – 24 = (x – 4)(x + 6)
\]