Simplifying algebraic expression with fraction of polynomials

Question

I cannot seem to find a way to simplify the following expression: $$\frac{9{a}^{4}-{a}^{2}{ b }^{ 4 } +16 { b }^{ 8 } }{ 3 { a }^{ 2 } -5a { b }^{ 2 } +4 { b }^{ 4 } }$$ I have tried factoring by $a$ and rewriting the division as multiplication with inverse.

Answer 1

A straightforward solution, simply use long division.

\begin{array}{rrrrrrr} & & 3a^2 &+5a b^2&+4b^4&&\\ \hline 3a^2-5ab^2+4b^4 & | & 9a^4 & & -a^2b^4 &&+ 16 b^8\\ &&-9a^4&+15a^3b^2&-12a^2b^4\\ \hline &&&15a^3b^2&-13a^2b^4&&+16b^8\\ &&&-15a^3b^2&+25a^2b^4&-20ab^6\\ \hline &&&&12a^2b^4&-20a b^6&+16b^8\\ &&&&-12a^2b^4&+20ab^6&-16b^8\\ \hline &&&&&&0 \end{array}

Answer 2

The numerator $9a^4 - a^2 b^4 + 16b^8$ factorises as $(3a^2+4b^4+5ab^2)(3a^2+4b^4−5ab^2)$.

Thus, when the $3a^2+4b^4−5ab^2\not=0$, $$\frac{9a^4 - a^2 b^4 + 16b^8}{3a^2+4b^4−5ab^2}= 3a^2+4b^4+5ab^2$$