An elastic string of unstretched length L and force constant k is...

In the string elastic force is conservative in nature. $\therefore \mathrm{W}=-\Delta \mathrm{U}$ Work done by elastic force of string, $$\begin{aligned}\mathrm{W}&=-\left(\mathrm{U}_{\mathrm{F}}-\mathrm{U}_{\mathrm{i}}\right)=\mathrm{U}_{\mathrm{i}}-\mathrm{U}_{\mathrm{F}}\\ \mathrm{W}&=\frac{1}{2} \mathrm{k} \mathrm{x}^{2}-\frac{\mathrm{k}}{2}(\mathrm{x}+\mathrm{y})^{2}\\ &=\frac{1}{2} \mathrm{k} \mathrm{x}^{2}-\frac{1}{2} \mathrm{k} \left(\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{xy}\right)\\ &=\frac{1}{2} \mathrm{k} \mathrm{x}^{2}-\frac{1}{2} \mathrm{k} \mathrm{y}^{2}-\frac{1}{2} \mathrm{kx}^{2}-\frac{1}{2} \mathrm{k}(2 \mathrm{xy})\\ &=-\mathrm{k} \mathrm{x} \mathrm{y}-\frac{1}{2} \mathrm{k} \mathrm{y}^{2} \end{aligned}$$Therefore,$the$work done against elastic force \mathrm{W}_{\text{external }}=-\mathrm{W}=\frac{\mathrm{K} y}{2}(2 \mathrm{x}+\mathrm{y})