Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Question

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?

Answer 1

Simplifying something is to some extent not a well-defined problem, because there is a valid (and not canonically answerable) question of what actually constitutes a simpler expression for something. For instance, in the problem at hand, you can actually write it as a single fraction by obtaining a common denominator as follows:

$\frac{5x}{(x+2)(x-3)}+\frac{4}{(x+2)^2}=\frac{5x(x+2)}{(x+2)^2(x-3)}+\frac{4(x-3)}{(x+2)^2(x-3)}=\frac{5x(x+2)+4(x-3)}{(x+2)^2(x-3)}$

One can simplify the numerator some, obtaining:

$\frac{5x^2+10x+4x-12}{(x+2)^2(x-3)}=\frac{5x^2+14x-12}{(x+2)^2(x-3)}$

However the numerator does not factor over the rationals ($\mathbb{Q}$), hence it cannot be 'simplified' any further. Now for the interesting question: Is this form really simpler than the one from your answer book? That is a really matter of opinion.

Edit: To make this point through example, consider $f(x)$ equals the expression given in the problem, and suppose you were asked to graph $f$. In that case, the form I derived is easier to use because of the theorems pertaining to the graph of a rational function given in (some?) College Algebra courses. However if you were instead asked to find an antiderivative of $f$ (as in Calculus), then it would be better to leave it in the form given in the answer book because that expression is much more readily integrated via $u$-subs and/or partial fractions.