We are given:
\(\displaystyle{\left(\frac{{4}}{{{k}-{1}}}\right)}-{\left(\frac{{1}}{{{k}+{4}}}\right)}={0}\)

Add \(\displaystyle\frac{{1}}{{{k}+{4}}}\) to both sides: \(\displaystyle{\left(\frac{{4}}{{{k}-{1}}}\right)}={\left(\frac{{1}}{{{k}+{4}}}\right)}\)

Cross multiply: 4(k+4)=1(k-1)

4k+16=k-1

Subtract 16 from both sides: 4k=k-17

Subtract k from both sides: 3k=-17

Divide both sides by 3: \(\displaystyle{k}=-{\left(\frac{{17}}{{3}}\right)}\)

Add \(\displaystyle\frac{{1}}{{{k}+{4}}}\) to both sides: \(\displaystyle{\left(\frac{{4}}{{{k}-{1}}}\right)}={\left(\frac{{1}}{{{k}+{4}}}\right)}\)

Cross multiply: 4(k+4)=1(k-1)

4k+16=k-1

Subtract 16 from both sides: 4k=k-17

Subtract k from both sides: 3k=-17

Divide both sides by 3: \(\displaystyle{k}=-{\left(\frac{{17}}{{3}}\right)}\)