Impulse response of a system in z domain
You are right.

In the solution, the second line which performs the long division

$$ H(z) = -4 + \frac{ 5 + \frac{7}{2} z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} } $$

is wrong and should be corrected as:

$$ H(z) = -4 + \frac{ 5 - 3 z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} } .$$

However, the partial fraction expansion at the following third line is correct; indicating there was a typo in the second line. Finally, the impulse response (which is based on line three) is also correct.