1. Simplify
[tex]\[ \frac{3}{x^2+14x+48} \div \frac{3}{10x+60} \][/tex]

A. [tex]\(\frac{10}{x+8}\)[/tex]

B. [tex]\(\frac{4}{x-2}\)[/tex]

C. [tex]\(\frac{3x}{x+1}\)[/tex]

D. [tex]\(x-4\)[/tex]



Answer :

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.