If `x gt 0,` which of the following is equivalent to `(2)/((1)/(x +6)+(1)/(x+2))` ?

Options

A. `x ^(2) + 8x+12`
B. `(x+4)/(x ^(2) + 8x + 12)`
C. `2x +8`
D. `(x^(2)+ 8x + 12)/(x+4)`

Explanation

To simplify the expression \(\frac{2}{\frac{1}{x + 6} + \frac{1}{x + 2}}\), we will follow these steps:

### Step 1: Combine the fractions in the denominator
The expression in the denominator is \(\frac{1}{x + 6} + \frac{1}{x + 2}\). To combine these fractions, we need a common denominator. The common denominator will be \((x + 6)(x + 2)\).

\[ \frac{1}{x + 6} + \frac{1}{x + 2} = \frac{(x + 2) + (x + 6)}{(x + 6)(x + 2)} = \frac{2x + 8}{(x + 6)(x + 2)} \]

### Step 2: Substitute back into the original expression
Now substitute this back into the original expression:

\[ \frac{2}{\frac{2x + 8}{(x + 6)(x + 2)}} \]

### Step 3: Simplify the expression
Dividing by a fraction is the same as multiplying by its reciprocal:

\[ = 2 \cdot \frac{(x + 6)(x + 2)}{2x + 8} \]

### Step 4: Simplify further
Notice that \(2x + 8\) can be factored out:

\[ 2x + 8 = 2(x + 4) \]

So we can rewrite our expression as:

\[ = 2 \cdot \frac{(x + 6)(x + 2)}{2(x + 4)} = \frac{(x + 6)(x + 2)}{x + 4} \]

### Step 5: Final expression
Thus, the expression simplifies to:

\[ \frac{(x + 6)(x + 2)}{x + 4} \]

### Conclusion
The equivalent expression for \(\frac{2}{\frac{1}{x + 6} + \frac{1}{x + 2}}\) is \(\frac{(x + 6)(x + 2)}{x + 4}\).

---