Algebraic operations between two expressions: how to include a constant outside a fraction into the numerator of that fraction

Question

I'm looking at a problem where the derivative of a function is defined as:

$$f'(x) = a-\frac{3x^2(1+x^2)-2x(x^3)}{(1+x^2)^2}$$

Then at the next line, the expression is expressed with the numerator as a quadratic function of $x^2$:

$$\frac{a+(2a-3)x^2+(a-1)x^4}{(1+x^2)^2}$$

But it is not clear what the steps are to get to this second expression.

I can see that the $f'(x)$ can be written as:

$$a-\frac{3x^2+x^4}{(1+x^2)^2}$$

But then to include $a$ into the numerator and come to the second expression has got me stuck...

Anyone suggestions or a hint?

Answer 1

$$a=\frac{a(1+x^2)^2}{(1+x^2)^2}=\frac{a+2ax^2+ax^4}{(1+x^2)^2}.$$

Answer 2

You have\begin{align}f'(x)&= a-\frac{3x^2(1+x^2)-2x(x^3)}{(1+x^2)^2}\\&=\frac{a(1+2x^2+x^4)-3x^2-3x^4+2x^4}{(1+x^2)^2}\\&=\frac{a+(2a-3)x^2+(a-1)x^2}{(1+x^2)^2}.\end{align}