Question

Simplify the complex rational expression. Type your answer in simplest form, multiplying any factors you may have in the numerator or denominator. When typing your answers, type your terms with variables in descending power and in alphabetical order without any spaces between your characters. If needed use the carrot key ^ (press shift and 6) to indicate an exponent.\frac{\left(\frac{2c}{c+2}+\frac{c-1}{c+1}\right)}{\frac{\left(2c+1\right)}{c+1}}

Answer 1

We will simplify the complex expression shown below:

`((2c)/(c+2)+(c-1)/(c+1))/((2c+1)/(c+1))`

The simplification process is shown below >>>>

`\begin{gathered} ((2c)/(c+2)+(c-1)/(c+1))/((2c+1)/(c+1)) \\ =((2c(c+1)+(c-1)(c+2))/((c+2)(c+1)))/((2c+1)/(c+1)) \\ =((2c^2+2c+c^2+2c-c-2)/((c+2)(c+1)))/((2c+1)/(c+1)) \\ =((3c^2+3c-2)/((c+2)(c+1)))/((2c+1)/(c+1)) \\ =(3c^2+3c-2)/((c+2)(c+1))*(c+1)/(2c+1) \\ =\frac{3c^2+3c-2}{(c+2)\cancel{c+1}}*\frac{\cancel{c+1}}{2c+1} \\ =(3c^2+3c-2)/((c+2)(2c+1)) \end{gathered}`

If we multiply out the denominator, we have:

`\begin{gathered} (3c^2+3c-2)/((c+2)(2c+1)) \\ =(3c^2+3c-2)/(2c^2+c+4c+2) \\ =(3c^2+3c-2)/(2c^2+5c+2) \end{gathered}`Answer

The numerator is

`3c^2+3c-2`

The denominator is

`2c^2+5c+2`