Problem 53

(1) Analysis
The problem involves the division operation of complex numbers. We can solve it according to the rules of operation.

Solution
$\frac{2-i}{3+4i} = \frac{(2-i)(3-4i)}{(3+4i)(3-4i)} = \frac{2-11i}{25} = \frac{2}{25} - \frac{11}{25}i$,

Hence, the answer is $\boxed{\frac{2}{25} - \frac{11}{25}i}$.

(2) Analysis
This problem involves the operation of derivatives. First, find the derivative function, then substitute $x=2$ to find the expression, and then solve the problem.

Solution
$f'(x)=2x+3f'(2)$, substituting $x=2$ gives $f'(2)=-2$, hence $f'(x)=2x-6$,

$1+f'(1)=1+2-6=-3$.

Hence, the answer is $\boxed{-3}$.