*x*2 + 14*x* + 45/*x*2 + 16*x* + 63

To find the domain of the expression \(\frac{x^2 + 14x + 45}{x^2 + 16x + 63}\), we need to determine the values of \(x\) that make the denominator equal to zero, as these values will make the expression undefined.

### Step 1: Set the denominator equal to zero.
The denominator is \(x^2 + 16x + 63\). We set this equal to zero:

\[
x^2 + 16x + 63 = 0
\]

### Step 2: Factor the quadratic equation.
We need to factor the quadratic equation. We are looking for two numbers that multiply to \(63\) and add to \(16\).

The factors are \(7\) and \(9\), so we can factor the quadratic as:

\[
(x + 7)(x + 9) = 0
\]

### Step 3: Solve for \(x\).
Set each factor equal to zero:

\[
x + 7 = 0 \quad \text{or} \quad x + 9 = 0
\]

\[
x = -7 \quad \text{or} \quad x = -9
\]

### Step 4: Determine the domain.
The expression is undefined at \(x = -7\) and \(x = -9\). Therefore, these values must be excluded from the domain.

### Final Answer:
The domain of the expression is all real numbers except \(x = -7\) and \(x = -9\). In interval notation, the domain is:

\[
(-\infty, -9) \cup (-9, -7) \cup (-7, \infty)
\]

Would you like any further details or have any questions?

Here are 5 related questions for practice:
1. What is the domain of the expression \(\frac{x^2 - 5x + 6}{x^2 + 7x + 10}\)?
2. How do you factor a quadratic expression?
3. What is the domain of \(\frac{x^2 + 3x + 2}{x^2 + 4x + 4}\)?
4. How can completing the square help in finding the roots of a quadratic equation?
5. Determine the domain of the expression \(\frac{x^2 + 10x + 25}{x^2 + 12x + 35}\).

**Tip:** When factoring quadratic expressions, always check your factor pairs to ensure they correctly match both the sum and the product needed to factor the quadratic equation.

Learn how to find the domain of the rational expression (x^2 + 14x + 45)/(x^2 + 16x + 63) with a step-by-step solution. Understand the quadratic formula and factoring quadratic equations. Suitable for high school students and above.

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