Let’s solve the given division problem step by step.

The problem is:

$\frac{x+1}{4x+12} \div \frac{x^2 + 3x + 2}{x^2 + 2x - 3}.$

Step 1: Rewrite as a multiplication

Division of fractions can be rewritten as multiplication by the reciprocal: $\frac{x+1}{4x+12} \times \frac{x^2 + 2x - 3}{x^2 + 3x + 2}.$

Step 2: Factorize all expressions

Factorize $4x + 12$:

$4x + 12 = 4(x + 3).$

Factorize $x^2 + 3x + 2$:

$x^2 + 3x + 2 = (x + 1)(x + 2).$

Factorize $x^2 + 2x - 3$:

$x^2 + 2x - 3 = (x + 3)(x - 1).$

So, the rewritten problem is: $\frac{x+1}{4(x+3)} \times \frac{(x+3)(x-1)}{(x+1)(x+2)}.$

Step 3: Simplify the expression

Cancel out common terms:

1. $x + 1$ in the numerator and denominator.
2. $x + 3$ in the numerator and denominator.

The simplified expression becomes: $\frac{1}{4} \times \frac{x - 1}{x + 2}.$

Step 4: Multiply the fractions

$\frac{1}{4} \times \frac{x - 1}{x + 2} = \frac{x - 1}{4(x + 2)}.$

Final Answer:

$\frac{x - 1}{4(x + 2)}.$

Would you like further explanation of any of these steps?

Related Questions:

1. How do you simplify rational expressions?
2. What are the steps to factorize quadratic expressions?
3. Why do we cancel out common terms in fractions?
4. How does reciprocal multiplication work in fraction division?
5. What are some examples of simplifying complex rational expressions?

Tip:

Always factorize all parts of the fraction completely before simplifying to avoid missing common factors!