factorise this quadratic x^2 + 9x + 20/x^2 + 22x + 30

To factorize the expression

\[
\frac{x^2 + 9x + 20}{x^2 + 22x + 30},
\]

we'll factorize both the numerator and the denominator separately.

### Step 1: Factorize the Numerator
The numerator is \( x^2 + 9x + 20 \).

We look for two numbers that multiply to \(20\) and add up to \(9\). These numbers are \(4\) and \(5\). So:

\[
x^2 + 9x + 20 = (x + 4)(x + 5)
\]

### Step 2: Factorize the Denominator
The denominator is \( x^2 + 22x + 30 \).

We look for two numbers that multiply to \(30\) and add up to \(22\). These numbers are \(2\) and \(15\). So:

\[
x^2 + 22x + 30 = (x + 2)(x + 15)
\]

### Step 3: Write the Factored Form
Now, we can write the factored form of the expression as:

\[
\frac{x^2 + 9x + 20}{x^2 + 22x + 30} = \frac{(x + 4)(x + 5)}{(x + 2)(x + 15)}
\]

This is the fully factored form. No further simplification can be done since there are no common factors between the numerator and the denominator.

Would you like more details, or do you have any questions?

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### Related Questions
1. How do you factor other types of quadratics, such as \( x^2 + bx + c \)?
2. What if the quadratic expression includes a leading coefficient other than 1, like \( ax^2 + bx + c \)?
3. Can you explain how to simplify rational expressions with multiple variables?
4. How do you find the roots of a quadratic equation?
5. What is the process to simplify expressions involving radicals?

#### Tip
Always check for common factors between the numerator and the denominator after factoring each component in rational expressions.

Explore how to factorize the quadratic expression x^2 + 9x + 20 / x^2 + 22x + 30 with a clear, step-by-step solution. This solution involves factoring the numerator and denominator of the rational expression, using key algebraic methods for quadratic expressions. Perfect for Grades 8-10 students.

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