Question

divide and simplify

x+4/x^2+2 divided by x^2-2/x-8

Answer 1

To simplify the expression \(\frac{\frac{x+4}{x^2+2}}{\frac{x^2-2}{x-8}}\), we can start by multiplying by the reciprocal of the second fraction.

This gives us:

\[
\frac{x+4}{x^2+2} \times \frac{x-8}{x^2-2}
\]

Now, we have:

\[
\frac{(x+4)(x-8)}{(x^2+2)(x^2-2)}
\]

Next, we simplify the denominator:

\[
x^2 - 2 \text{ can be factored using the difference of squares: } (x-\sqrt{2})(x+\sqrt{2})
\]

So, we rewrite the denominator:

\[
(x^2 + 2)(x^2 - 2) = (x^2 + 2)((x - \sqrt{2})(x + \sqrt{2}))
\]

Now our expression looks like:

\[
\frac{(x+4)(x-8)}{(x^2+2)(x - \sqrt{2})(x + \sqrt{2})}
\]

We can stop here unless the question specifies a further simplification such as factorization or cancellation.

The final expression, before substituting any specific values for \(x\) or expanding, is:

\[
\frac{(x+4)(x-8)}{(x^2+2)(x-\sqrt{2})(x+\sqrt{2})}
\]

This is the most simplified form we can get without numeric values or specific factorization of the polynomials in the numerator and denominator.