To find the greatest common factor (GCF) of the given rational expression:
[tex]\[
\frac{9x + 45}{x^2 + 9x + 20}
\][/tex]
we need to first factor both the numerator and the denominator.
### Step 1: Factor the Numerator
The numerator is [tex]\(9x + 45\)[/tex].
Notice that both terms have a common factor:
[tex]\[
9x + 45 = 9(x + 5)
\][/tex]
### Step 2: Factor the Denominator
The denominator is [tex]\(x^2 + 9x + 20\)[/tex].
To factor this quadratic expression, we look for two numbers that multiply to 20 (the constant term) and add up to 9 (the coefficient of [tex]\(x\)[/tex]).
The numbers that fulfill these requirements are 4 and 5. Therefore, we can factor the quadratic as:
[tex]\[
x^2 + 9x + 20 = (x + 4)(x + 5)
\][/tex]
### Step 3: Identify the Common Factor
Now that we have factored both the numerator and the denominator, we can see the expressions in their factored forms:
The numerator is [tex]\(9(x + 5)\)[/tex] and the denominator is [tex]\((x + 4)(x + 5)\)[/tex].
Both the numerator and the denominator share a common factor of [tex]\(x + 5\)[/tex].
### Step 4: Conclusion
Hence, the greatest common factor (GCF) of the numerator and the denominator is:
[tex]\[
\boxed{x + 5}
\][/tex]
