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Factoring quadratics is a crucial skill in solving rational equations. A quadratic is an equation of the form \(ax^2 + bx + c\). Factoring involves rewriting this expression as a product of two binomials. In the given exercise, the quadratic \(a^2 - 4a - 21\) is present. To factor it, look for two numbers that multiply to give \(-21\) (the constant term) and add up to \(-4\) (the linear coefficient).

Those numbers are \(-7\) and \(3\). Consequently, the quadratic factors as \((a-7)(a+3)\). Factoring simplifies the rational equation and makes it easier to find the least common denominator. Remember to always check for factorization first whenever a quadratic is involved in equations, as it often leads the way to simpler forms.

The least common denominator (LCD) is essential for simplifying and solving rational equations involving fractions. The LCD of several fractions ensures that all terms in the equation have the same denominator, allowing them to be combined or compared directly. In the provided exercise, the denominators are \((a+3)\), \((a^2 - 4a - 21)\), and \((a-7)\).

After factoring \(a^2 - 4a - 21\) to \((a-7)(a+3)\), it's clear that the LCD is \((a-7)(a+3)\). To eliminate the denominators, multiply each term of the equation by the LCD. This step transforms the rational equation into a linear equation, which is more straightforward to solve. Always identify the LCD by factoring any quadratic expressions first, as it simplifies the solution process.

After using the LCD to clear the fractions, you'll often end up with a linear equation. Solving linear equations involves isolating the variable on one side of the equation. Consider the linear equation obtained from the original problem: \(3(a-7) + 14 = 5(a+3)\).

Simplify both sides by distributing and combining like terms to get \(3a - 21 + 14 = 5a + 15\). Then, combine constants and variables separately: rearrange terms to isolate \(a\). This involves moving all terms containing \(a\) to one side and constants to the other. For this equation, you'll end up with \(2a = -8\).

Finally, divide both sides by the coefficient of \(a\) to solve for the variable. This yields \(a = -4\). Understanding the step-by-step process of solving linear equations is essential, as it provides the foundation for solving more complex algebraic problems successfully.