using $ 2 \cos ^{2} A=1+\cos 2 A$

$x=3 \sin 100 t+4 \cos 100 t+4$

$\frac{d x}{d t}=300 \cos 100 t-400 \sin 100 t$

$\frac{d^{2}(x-4)}{d t^{2}}=-30000 \sin 100 t+40000 \cos 100 t=-100(3 \sin 100 t+4 \cos 100 t)=$

$-100(x-4)$

so it is a $SHM.$

Amplitude of the motion, $A=\sqrt{3^{2}+4^{2}}=5$ units

for max distance, $\frac{d x}{d t}=300 \cos 100 t-400 \sin 100 t=0 \Rightarrow \tan 100 t=\frac{3}{4}$

$\therefore \sin 100 t=\frac{3}{5}, \cos 100 t=\frac{4}{5}$

now, $x_{\max }=3 \times \frac{3}{5}+4 \times \frac{4}{5}+4=5+4=9$ units