Express the following in the lowest terms.
`(x^(2)+6x+9)/(x^(2)-9)`

Answer

To express the given expression \((x^2 + 6x + 9)/(x^2 - 9)\) in its lowest terms, we will follow these steps:

### Step 1: Factor the numerator and denominator
The numerator is \(x^2 + 6x + 9\). We can factor this expression. We look for two numbers that multiply to \(9\) (the constant term) and add up to \(6\) (the coefficient of \(x\)). The numbers \(3\) and \(3\) satisfy these conditions.

Thus, we can factor the numerator as:
\[ x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 \]

The denominator is \(x^2 - 9\). This is a difference of squares, which can be factored as:
\[ x^2 - 9 = (x - 3)(x + 3) \]

### Step 2: Rewrite the expression with the factored forms
Now we can rewrite the original expression using the factored forms:
\[ \frac{x^2 + 6x + 9}{x^2 - 9} = \frac{(x + 3)^2}{(x - 3)(x + 3)} \]

### Step 3: Simplify the expression
Next, we can simplify the expression by canceling out the common factors in the numerator and the denominator. The common factor is \((x + 3)\):
\[ \frac{(x + 3)^2}{(x - 3)(x + 3)} = \frac{x + 3}{x - 3} \]

### Final Answer
Thus, the expression in its lowest terms is:
\[ \frac{x + 3}{x - 3} \]

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