In this article, we’ll explore how to simplify the complex expression . This involves understanding how to work with complex fractions, factor quadratic expressions, and simplify the result step by step.

## Step 1: Rewrite the Expression

We begin by recognizing that the given expression is a quotient of two terms. The expression can be rewritten as:

This shows that we are dividing the fraction .

## Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression in the denominator, . To factor a quadratic of the form , we look for two numbers that multiply to (12) (the constant term) and add to (7) (the coefficient of the middle term).

The two numbers are (3) and (4) because:

Thus, we can factor as:

## Step 3: Substitute the Factored Form

We now substitute this factored form of into the expression:

## Step 4: Simplify the Expression

Now that we have the expression &, we can apply the rule for dividing by a fraction. Dividing by an expression is the same as multiplying by its reciprocal. This gives:

At this point, we can cancel the common factor of (t+3) from both the numerator and the denominator, leaving us with:

## Final Answer

The simplified quotient is:

### Conclusion

The expression simplifies to . This result was obtained by factoring the quadratic expression in the denominator, applying the rule for dividing fractions, and simplifying common terms. The final answer is a much simpler expression with a single denominator, showing how powerful factoring and simplification can be in algebra.