Answer:
[tex]\dfrac{1}{x^2+4x+3}[/tex]
Step-by-step explanation:
You have asked for the simpilfied form of ...
[tex]x-\dfrac{2}{x^2}+x-\dfrac{6}{x^2}+5x+\dfrac{4}{x}+4[/tex]
That would be ...
[tex]\dfrac{7x^3+4x^2+4x-4}{x^2}[/tex]
We suspect that's not what you meant. Parentheses are required for grouping when you write math expressions in text form. They work best using math symbols instead of words. We think you mean
... ((x -2)/(x^2 +x -6))/((x^2 +5x +4)/(x +4))
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Maybe you want to simplify ...
[tex]\displaystyle\frac{\left(\frac{x-2}{x^2+x-6}\right)}{\left(\frac{x^2+5x+4}{x+4}\right)}=\frac{(x-2)(x+4)}{(x-2)(x+3)(x+4)(x+1)}\\\\=\frac{1}{(x+1)(x+3)}=\frac{1}{x^2+4x+3}[/tex]
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Comment on simplifying rational expressions
Division of fractions works the same whether you're working with numbers or polynomials (or anything else). Dividing by something is the same as multiplying by its inverse (reciprocal).
(a/b)/(c/d) = (a/b)·(d/c)
I learned this as "invert and multiply". I've recently seen it referred to as "copy dot flip", meaning you copy the numerator, use a dot symbol to indicate multipication, then flip the denominator (make its reciprocal) to become what you're multiplying by.
