Precalc coursework - Is this an error? I am enrolled in a precalc math course at a university studying online. I haven't studied math in over 35 years so I'm a bit rusty. Why would my online math coursework have errors? I end up torn between feeling like I must just not understand something and certainty that there is an error in my professor's notes. I turn to you for confirmation and hopefully confidence. 
I attach an image of the professor's solution, please could you confirm for me whether my belief that she is missing an X factor in her work and she then continues solving the equation incorrectly. Who is right?


You are right. There is a missing $x.$ The final result should be $\dfrac{x+4}{2x^2}.$ Note that
$$\begin{align}\frac{1}{x+3}\left(\frac{x+7}{2x}+\frac{6}{x^2}\right)& = \frac{1}{x+3}\left(\frac{(x+7)\cdot x}{2x\cdot x}+\frac{6\cdot 2}{x^2\cdot 2}\right)\\ &= \frac{1}{x+3}\left(\frac{x^2+7 x}{2x^2}+\frac{12}{2x^2}\right) \\ &= \frac{1}{x+3}\frac{x^2+7 x+12}{2x^2} \\ &= \frac{x^2+7 x+12}{2x^2(x+3)}  \\ &= \frac{(x+4)(x+3)}{2x^2(x+3)}\\ &= \frac{x+4}{2x^2}.\end{align}$$
