`((x + 5) (x^(2) + 7x + 10))/((x + 3) (x^(2) + 10x + 25))` can be simplifed in

Options

A. `(x + 2)/(x + 3)`
B. `(x - 2)/(x - 3)`
C. `(x + 5)/(x + 3)`
D. `(x + 3)/(x + 5)`

Answer

To simplify the expression \(\frac{(x + 5)(x^2 + 7x + 10)}{(x + 3)(x^2 + 10x + 25)}\), we will follow these steps:

### Step 1: Factor the Quadratic Expressions
First, we need to factor the quadratic expressions in the numerator and the denominator.

1. **Numerator: \(x^2 + 7x + 10\)**
- We need two numbers that multiply to \(10\) (the constant term) and add up to \(7\) (the coefficient of \(x\)). The numbers \(5\) and \(2\) satisfy this.
- Thus, we can factor it as:
\[ x^2 + 7x + 10 = (x + 5)(x + 2) \]

2. **Denominator: \(x^2 + 10x + 25\)**
- We need two numbers that multiply to \(25\) and add up to \(10\). The numbers \(5\) and \(5\) satisfy this.
- Thus, we can factor it as:
\[ x^2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)^2 \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression with the factored forms:
\[ \frac{(x + 5)(x + 5)(x + 2)}{(x + 3)(x + 5)(x + 5)} \]

### Step 3: Cancel Common Factors
Next, we can cancel the common factors in the numerator and denominator:
- The factor \((x + 5)\) appears in both the numerator and the denominator.

After canceling, we have:
\[ \frac{x + 2}{x + 3} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ \frac{x + 2}{x + 3} \]