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The difference of squares is a common pattern in algebra that makes certain expressions easier to factor and simplify. When you see an expression like \(a^2 - b^2\), you can quickly apply a formula to simplify it. This formula is \(a^2 - b^2 = (a - b)(a + b)\). It tells us that the difference between the squares of two terms is equal to the product of the sum and the difference of those terms. Recognizing this pattern saves you time and helps in simplifying rational expressions.
For example, in the given exercise \(a^2 - b^2\) is present in the numerator. By applying the difference of squares formula, we can factor this to get \((a - b)(a + b)\). This step is crucial for simplifying the rational expression further.

Factoring polynomials is a fundamental skill in algebra. It involves writing a polynomial as a product of its factors instead of a sum or difference of terms. In simpler terms, you break down a complex expression into simpler pieces that, when multiplied together, give you the original polynomial.
In our exercise, after identifying and applying the difference of squares to the numerator \(a^2 - b^2\), we see that it can be factored into \((a - b)(a + b)\). Factoring helps in comparing the numerator and the denominator of rational expressions to find any common factors. However, in this problem, the denominator \(a^2 + b^2\) can't be factored over the real numbers showing there are no common factors, thus the expression is already in its simplest form once factored.

The sum of squares, represented by \(a^2 + b^2\), is another common pattern in algebra. However, unlike the difference of squares, the sum of squares doesn't have a simple factoring formula over the real numbers. In fact, \(a^2 + b^2\) is considered prime in the set of real numbers because it cannot be expressed as a product of other real polynomials.
In our exercise, the denominator is a sum of squares \(a^2 + b^2\). Since it doesn't factor over the real numbers, we confirm there are no shared factors with the numerator. This is essential to verify that the rational expression remains in its lowest terms. Always remember this distinction between sums and differences of squares as it guides you on how to approach factoring in polynomial expressions.