5.1 Rational operations (complex fractions)

If [math]\displaystyle{\frac{2x+1-\frac{x-1}{2x+1}}{1-\frac{1}{2x+1}}}[/math] can be written as [math]\dfrac{ax^2+bx+c}{dx+e}[/math] where [math]a, b, c, d[/math], and [math]e[/math] are integers, what are [math]a, b, c, d[/math], and [math]e[/math]?

If [math]\displaystyle{\frac{1}{\frac{1}{x+1}+\frac{1}{x-1}}}[/math] can be written as [math]\dfrac{ax^2+bx+c}{dx+e}[/math] where [math]a, b, c, d[/math], and [math]e[/math] are integers, what are [math]a, b, c, d[/math], and [math]e[/math]?

Which of the following expressions is equivalent to [math]\left(\dfrac{x}{x+1}+x\right)^{-1}[/math]?

[math]\dfrac{\quad\frac{x-3}{x^2+6}\quad}{\frac{x^2-9}{x+3}}=\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials. If the leading coefficient of [math]q(x)[/math] is 1 and [math]q(x)[/math] is quadratic, what is [math]p(2025)[/math]?

If [math]\dfrac{\frac{3x+1}{x^2-x-12}+3}{\frac{3}{x-4}+\frac{4}{x+3}}[/math] can be written as [math]\dfrac{ax^2+bx+c}{dx+e}[/math] where [math]a, b, c, d[/math], and [math]e[/math] are integers, what are [math]a, b, c, d[/math], and [math]e[/math]?

[math]\dfrac{4-\frac{8x}{x^2+1}}{x-1}=\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials. If the leading coefficient of [math]q(x)[/math] is 1, what is the leading coefficient of [math]p(x)[/math]?

Which of the following expressions is equivalent to [math]\dfrac{\frac{1}{x+h+2}-\frac{1}{x+2}}{\frac{1}{h}}[/math]?

Which of the following is equivalent to [math]\displaystyle{\frac{\displaystyle{\frac{8}{7x}}}{\displaystyle{\frac{-4}{x+1}}}}[/math]?