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Factoring by grouping is an important technique to factor polynomials, especially when dealing with quadratic expressions. In this method, we arrange the terms into groups that can be factored easily. This helps in breaking down complex expressions into simpler, more manageable parts.

Quadratic expressions take the form of ax^2 + bx + c, where 'a,' 'b,' and 'c' are constants. In the given exercise, the expression is in terms of 'a' and 'b' instead of the usual 'x.' Here, the expression is a^{2} - a b - 12 b^{2}, with the coefficients being: a^{2} (leading term), -ab (middle term), and -12b^{2} (constant term). Understanding how to identify and work with these components is crucial to solving such problems.

Polynomial factoring involves expressing a polynomial as a product of its factors. For the given exercise, after identifying the quadratic expression and its coefficients, the next step is to use the AC method or trial and error to rewrite the expression in a form that can be grouped. The expression a^{2} - a b - 12 b^{2} is rewritten using -4ab and 3ab, resulting in: a^{2} - 4ab + 3ab - 12b^{2}.
The terms are then grouped and factored as follows: (a^{2} - 4ab) + (3ab - 12b^{2}). Factoring out the common terms within each group gives: a(a - 4b) + 3b(a - 4b). Finally, the common binomial factor (a - 4b) is factored out, resulting in: (a - 4b)(a + 3b). This shows the fully factored form of the original polynomial a^{2} - a b - 12 b^{2}.