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Factoring quadratics is a staple in algebra. Quadratic expressions are polynomials of the form ewline ewline \(ax^2 + bx + c\), where we need to factor them into simpler expressions. This process requires finding two binomials that multiply together to give the original quadratic. In the given exercise, we start with the numerator \(x^2 + 2x - 15\). We look for two numbers that multiply to \(-15\) (the constant term) and add to \(2\) (the coefficient of \(x\)). These numbers are 5 and -3. Hence, we can rewrite the numerator as \((x + 5)(x - 3)\). Similarly, for the denominator \(x^2 + 6x + 5\), the numbers 1 and 5 satisfy the conditions, hence we get \((x + 1)(x + 5)\). Facoring quadratics requires practice, but once you understand it, it becomes an easy and important tool in algebra.

Quadratic expressions come up often in algebra and are crucial for various operations. When we see quadratics like \(x^2 + 2x - 15\) or \(x^2 + 6x + 5\), we need to understand their structure and behavior. Each quadratic expression can be plotted as a parabola on a graph. The solutions, or roots, of the quadratic are the \(x\)-values where the parabola intersects the x-axis. To find these roots, factoring the quadratic expression is often one of the most straightforward methods. For example, by factoring \(x^2 + 2x - 15\), we get the factors \((x + 5)(x - 3)\), which imply the roots are \(-5\) and \(3\). Similarly, \(x^2 + 6x + 5\) factors to \((x + 1)\) and \((x + 5)\), implying roots of \(-1\) and \(5\). These factorizations simplify many algebraic operations, like rational expressions.

Algebraic simplification is the process of making an algebraic expression as simple as possible. After factoring both the numerator and the denominator in the given exercise, we obtain: ewline ewline \(\frac{(x + 5)(x - 3)}{(x + 1)(x + 5)}\). Here, you can see the common term \((x + 5)\) in both the numerator and the denominator. By canceling this common term, we simplify the expression to ewlineewline \(\frac{x - 3}{x + 1}\). This step not only makes the expression simpler but also easier to work with for subsequent operations. The key is to fully factor both the numerator and the denominator to identify these common terms. Always check for possible simplifications, as it makes algebra more manageable and helps in solving complex problems.