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When adding or subtracting fractions, it's important that they have a common denominator. The denominator is the bottom part of a fraction, and it tells you into how many equal parts the whole is divided. Without a common denominator, you cannot directly add or subtract fractions.
For this exercise, we have two fractions with different denominators: \(x^2+6x+9\) and \(x+3\). The key is to find a common denominator that both can be converted to. By factoring the quadratic expression \(x^2+6x+9\), we get \((x+3)^2\). This means that the common denominator for these fractions is \((x+3)^2\).
Understanding how to find a common denominator helps in dealing with fractions efficiently, especially when they involve polynomials. It ensures both fractions "speak the same language," making addition or subtraction possible.

Polynomial factorization is the process of breaking down a polynomial into simpler polynomials that can be multiplied to give the original polynomial. This is a key step in many algebraic operations, including finding common denominators.
In our exercise, we encountered the polynomial \(x^2+6x+9\). It appears complex but can actually be factored into \((x+3)^2\). To factor it, recognize it as a perfect square trinomial. The middle term, 6x, is twice the product of the square root of each other term. This knowledge can save time and effort when dealing with complex fractions.
This simplification not only aids in adding or subtracting fractions but also helps in deeper algebraic problems, making it easier to handle equations and functions.

Adding fractions involves aligning them under a common denominator and then combining the numerators. Once the denominators are the same, you can treat it like straightforward addition or subtraction.
In the given exercise, once each fraction was expressed with the common denominator \((x+3)^2\), we simply add the numerators: \(4\) and \(4(x+3)\). It's like adding pieces of the same type of pie! This becomes \(4 + 4(x+3)\), which simplifies to \(4 + 4x + 12\). That's how you arrive at the combined numerator \(4x+16\).
Having a grasp on fraction addition by focussing on the numerators after leveling the denominators will make you more comfortable in tackling rational expressions in algebra.