Simplify (x^4 - 8x^2 + 16) / (x^2 + 5x + 6)

Hey everyone! Today, we're diving into a fun algebraic problem: simplifying the expression (x^4 - 8x^2 + 16) / (x^2 + 5x + 6). This might look a bit intimidating at first, but don't worry, we'll break it down step by step. We aim to identify which of the given options—A. (x^2 + 4)(x - 2) / (x + 3), B. (x - 2)^2 / (x + 3), C. (x - 2)^2(x + 2) / (x + 3), or D. ((x^2 - 4)^2) / ((x + 3)^2)—is equivalent to our original expression. So, let’s put on our algebraic hats and get started!

Factoring the Numerator: Unveiling the Hidden Structure

Our journey begins with the numerator: x^4 - 8x^2 + 16. The key to simplifying expressions like this lies in factoring. Factoring is like reverse multiplication; we're trying to find the expressions that multiply together to give us our original expression. When we gaze upon x^4 - 8x^2 + 16, it bears a striking resemblance to a perfect square trinomial. Remember the formula: (a - b)^2 = a^2 - 2ab + b^2? If we let a = x^2 and b = 4, we can see how this fits perfectly. Let's explore this further to ensure we grasp the essence of perfect square trinomials.

Recognizing the pattern of a perfect square trinomial is crucial. In our case, x^4 is (x²⁾2, and 16 is 4^2. The middle term, -8x^2, is -2 * x^2 * 4, which confirms our perfect square trinomial hunch. This recognition allows us to rewrite the numerator as (x^2 - 4)^2. But wait, there's more! We're not done yet. Inside the parentheses, we have another familiar friend: x^2 - 4. This is a classic difference of squares, which can be factored even further. The difference of squares formula is a^2 - b^2 = (a - b)(a + b). Applying this to x^2 - 4, where a = x and b = 2, we get (x - 2)(x + 2). Now, substituting this back into our expression, we have ((x - 2)(x + 2))^2. Remember, squaring a product means squaring each factor, so this becomes (x - 2)^2 * (x + 2)^2. See how factoring unveils the hidden structure? We've transformed a seemingly complex expression into a product of simpler terms, making it much easier to work with. Mastering these factoring techniques is fundamental in algebra, enabling us to simplify expressions, solve equations, and understand the underlying relationships between mathematical concepts. Keep practicing, and you'll become a factoring pro in no time!

Taming the Denominator: Factoring x^2 + 5x + 6

Now, let's shift our focus to the denominator: x^2 + 5x + 6. This is a quadratic expression, and like the numerator, it can be simplified through factoring. Factoring a quadratic involves finding two binomials (expressions with two terms) that multiply together to give us the original quadratic. So, how do we do this? The general form of a quadratic expression is ax^2 + bx + c, where a, b, and c are constants. In our case, a = 1, b = 5, and c = 6. The trick is to find two numbers that add up to b (which is 5) and multiply to c (which is 6). These numbers will be the constants in our binomial factors. Let's think about the factors of 6: we have 1 and 6, 2 and 3. Which pair adds up to 5? That's right, 2 and 3! Now we can rewrite the quadratic as (x + 2)(x + 3). Notice how the 2 and 3 directly become the constants in our binomials. This works because when we multiply (x + 2)(x + 3), we get x^2 + 3x + 2x + 6, which simplifies to x^2 + 5x + 6. It's like piecing together a puzzle! Factoring quadratics is a fundamental skill in algebra, and there are many different techniques you can use. This method, known as the "find two numbers" approach, is particularly helpful for simpler quadratics where the leading coefficient (a) is 1. As you encounter more complex quadratics, you might need to explore other methods like the quadratic formula or completing the square. But for now, mastering this basic factoring technique will take you a long way in simplifying expressions and solving equations. Remember, practice makes perfect, so keep working on those factoring skills!

Putting It All Together: Simplifying the Expression

Okay, we've conquered the numerator and the denominator separately. Now comes the fun part: putting it all together and simplifying the entire expression. We started with (x^4 - 8x^2 + 16) / (x^2 + 5x + 6). We factored the numerator into (x - 2)^2 * (x + 2)^2 and the denominator into (x + 2)(x + 3). So, our expression now looks like this: [(x - 2)^2 * (x + 2)^2] / [(x + 2)(x + 3)]. The beauty of factoring is that it allows us to see common factors in the numerator and denominator, which we can then cancel out. It's like simplifying a fraction by dividing both the top and bottom by the same number. Do you spot any common factors in our expression? Excellent! We have (x + 2)^2 in the numerator and (x + 2) in the denominator. This means we can cancel out one (x + 2) from both the top and the bottom. Remember, (x + 2)^2 means (x + 2) * (x + 2), so we're essentially dividing both the numerator and denominator by (x + 2). After canceling, our expression becomes [(x - 2)^2 * (x + 2)] / (x + 3). And that's it! We've simplified our expression as much as possible. See how factoring and canceling common factors can make a complex expression much more manageable? This process is a cornerstone of algebraic manipulation, allowing us to solve equations, graph functions, and understand the relationships between different mathematical expressions. Keep practicing these techniques, and you'll be simplifying expressions like a pro in no time!

Identifying the Equivalent Expression: Choosing the Correct Answer

Now that we've simplified our expression to [(x - 2)^2 * (x + 2)] / (x + 3), let's circle back to our original question. We were asked to identify which of the following options is equivalent to our starting expression: A. (x^2 + 4)(x - 2) / (x + 3), B. (x - 2)^2 / (x + 3), C. (x - 2)^2(x + 2) / (x + 3), or D. ((x^2 - 4)^2) / ((x + 3)^2). We've done all the hard work of simplifying, so this should be the easy part! Just compare our simplified expression, [(x - 2)^2 * (x + 2)] / (x + 3), with the options provided. Which one matches exactly? Bingo! Option C, (x - 2)^2(x + 2) / (x + 3), is a perfect match. Therefore, option C is the equivalent expression. This final step highlights the power of simplification. By breaking down a complex expression into its simplest form, we were able to easily identify the correct answer. This not only saves us time but also reduces the chances of making errors. When tackling similar problems, remember the importance of factoring, canceling common factors, and simplifying as much as possible. These techniques are your allies in the world of algebra, helping you navigate even the trickiest expressions with confidence. So, keep practicing, keep simplifying, and keep acing those math problems!

Therefore, the answer is C. (x - 2)^2(x + 2) / (x + 3).