Rational expression in lowest term - question from Euler’s element of algebra

Question

How do I convert this expression to lowest form?

$$\dfrac{5a^5+10a^4x+5a^3x^2}{a^3x+2a^2x^2+2ax^3+x^4}$$

Answer 1

Hint: factor the numerator, first, which should give a hint of what to look for in the denominator:

$$\require{cancel}\; \dfrac{5a^3(a^2+2ax+x^2)}{x(a^3+2a^2x+2ax^2+x^3)} = \dfrac{5a^3(a+x)^2}{x(a + x) (a^2 + a x + x^2)} $$