Given that `(x^2 + 6x + 9)/(x + 3) = 7`, what is x?

Answer

To solve the equation \(\frac{x^2 + 6x + 9}{x + 3} = 7\), we will follow these steps:

### Step 1: Simplify the left-hand side
The expression \(x^2 + 6x + 9\) can be factored. Notice that it is a perfect square trinomial:
\[ x^2 + 6x + 9 = (x + 3)^2 \]
Thus, we can rewrite the equation as:
\[ \frac{(x + 3)^2}{x + 3} = 7 \]

### Step 2: Cancel the common terms
Assuming \(x + 3 \neq 0\) (which means \(x \neq -3\)), we can simplify the left-hand side:
\[ x + 3 = 7 \]

### Step 3: Solve for \(x\)
Now, we can solve for \(x\) by isolating it:
\[ x + 3 = 7 \implies x = 7 - 3 \]
\[ x = 4 \]

### Step 4: Check for extraneous solutions
Since we canceled \(x + 3\) in our simplification, we need to ensure that our solution does not make the denominator zero. We check:
\[ x + 3 = 4 + 3 = 7 \neq 0 \]
Thus, our solution is valid.

### Final Answer
The value of \(x\) is:
\[ \boxed{4} \]

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