To simplify the expression
[tex]\[
\frac{3}{x^2+14x+48} \div \frac{3}{10x+60},
\][/tex]
we follow these steps:
### Step 1: Write the Division as Multiplication
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the division as:
[tex]\[
\frac{3}{x^2+14x+48} \times \frac{10x+60}{3}.
\][/tex]
### Step 2: Simplify the Expression
At this stage, notice that the [tex]\( \frac{3}{3} \)[/tex] cancels out, simplifying our expression:
[tex]\[
\frac{10x+60}{x^2+14x+48}.
\][/tex]
### Step 3: Factor the Denominator
Now, we need to factor the quadratic expression in the denominator. The expression [tex]\( x^2 + 14x + 48 \)[/tex] factors as:
[tex]\[
x^2 + 14x + 48 = (x + 6)(x + 8).
\][/tex]
So, our expression now looks like:
[tex]\[
\frac{10x + 60}{(x + 6)(x + 8)}.
\][/tex]
### Step 4: Factor the Numerator
Next, we factor the numerator [tex]\( 10x + 60 \)[/tex]:
[tex]\[
10x + 60 = 10(x + 6).
\][/tex]
So, the expression becomes:
[tex]\[
\frac{10(x + 6)}{(x + 6)(x + 8)}.
\][/tex]
### Step 5: Cancel Common Factors
We note that [tex]\( x + 6 \)[/tex] appears in both the numerator and the denominator, so they cancel each other out, leaving us with:
[tex]\[
\frac{10}{x + 8}.
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{10}{x + 8}}.
\][/tex]
This corresponds to option A.
