If `(x^(2)+5x+6)/(x+2)=12`, what is the value of x?

Options

A. `-2`
B. `2`
C. `3`
D. `9`

Answer

To solve the equation \(\frac{x^2 + 5x + 6}{x + 2} = 12\), we will follow these steps:

### Step 1: Factor the numerator
The numerator \(x^2 + 5x + 6\) can be factored. We need two numbers that multiply to \(6\) (the constant term) and add up to \(5\) (the coefficient of \(x\)). The numbers \(2\) and \(3\) satisfy this condition.

Thus, we can write:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]

### Step 2: Rewrite the equation
Now we can substitute the factored form back into the equation:
\[ \frac{(x + 2)(x + 3)}{x + 2} = 12 \]

### Step 3: Cancel the common terms
Since \(x + 2\) is in both the numerator and the denominator, we can cancel it (as long as \(x \neq -2\)):
\[ x + 3 = 12 \]

### Step 4: Solve for \(x\)
Now, we will solve for \(x\) by subtracting \(3\) from both sides:
\[ x = 12 - 3 \]
\[ x = 9 \]

### Final Answer
The value of \(x\) is \(9\).

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