Question

Simplify the following expression:
x^2+14x+40/x^2−2x−24





The simplified form is




What are the restrictions on the variable?
(Type only the numbers and separate your answers with commas, as needed.)

Answer 1

To simplify the expression \(\frac{x^2+14x+40}{x^2-2x-24}\), we will first factor both the numerator and the denominator.

**Step 1: Factor the numerator \(x^2+14x+40\).**

We need two numbers that multiply to \(40\) (the constant term) and add up to \(14\) (the coefficient of \(x\)). The numbers \(10\) and \(4\) work:

\[
x^2 + 14x + 40 = (x + 10)(x + 4)
\]

**Step 2: Factor the denominator \(x^2-2x-24\).**

We need two numbers that multiply to \(-24\) and add up to \(-2\). The numbers \(4\) and \(-6\) work:

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

**Step 3: Rewrite the expression using the factors.**

Now our expression looks like this:

\[
\frac{(x+10)(x+4)}{(x-6)(x+4)}
\]

**Step 4: Simplify the expression.**

We can cancel the common factor \((x + 4)\):

\[
\frac{(x+10)\cancel{(x+4)}}{(x-6)\cancel{(x+4)}} = \frac{x + 10}{x - 6}, \quad (x + 4 \neq 0)
\]

**Final Result:**
The simplified form of the expression is:

\[
\frac{x + 10}{x - 6}
\]

**Step 5: Identify restrictions on the variable.**

The expression is undefined when the denominator \(x - 6\) is zero or when \(x + 4\) is zero (since we canceled it out). Thus, we need to set these expressions to zero:

1. \(x - 6 = 0\) → \(x = 6\)
2. \(x + 4 = 0\) → \(x = -4\)

The restrictions on the variable \(x\) are:

\[
-4, 6
\]

**Final Answers:**

The simplified form is \(\frac{x + 10}{x - 6}\).
The restrictions on the variable are:

\[
-4, 6
\]