1：Experimental Design and Method Selection:

The nanoparticles were synthesized using the hydrothermal method to achieve good crystallinity and phase purity. Characterization techniques included XRD, FTIR, FESEM, TEM, UV-Vis spectroscopy, VSM, and P-E loop tracer to analyze various properties.

2：Sample Selection and Data Sources:

Samples with x = 0.2, 0.4, and 0.6 were prepared using precursors from Sigma-Aldrich (99.8% purity).

3：2, 4, and 6 were prepared using precursors from Sigma-Aldrich (8% purity). List of Experimental Equipment and Materials:

3. List of Experimental Equipment and Materials: Precursors (La(NO3)2, Ba(NO3)2·6H2O, Cu(NO3)2·6H2O, Fe(NO3)3·9H2O, NaOH), autoclave, hot-air oven, X-ray diffractometer (Bruker, XRD, CuKα, λ =

4：15406 nm), Transmission Electron Microscope (TEM:

Tecnai G20, FEI, USA), Field-emission Scanning Electron Microscope (Ultra 55 FE-SEM Carl Zeiss), FT-IR spectrophotometer (IR affinity-1, Shimadzu), UV-Visible spectrophotometer (JASCO V-670 PC), Vibrating Sample Magnetometer (EV-7 VSM with Max. applied field +15kOe), P-E loop tracer (Marine India Ltd.).

5：Experimental Procedures and Operational Workflow:

Precursors were mixed in distilled water, NaOH added to pH 12, transferred to autoclave, heated at 150°C for 8 h, washed and dried, then heated at 500 K to remove secondary phases. Characterization was performed on the resulting powder.

6：Data Analysis Methods:

XRD data analyzed using Debye Scherrer formula and Williamson-Hall plot; FTIR spectra interpreted for cation sites; optical bandgap calculated from UV-Vis data; magnetic parameters derived from M-H loops; ferroelectric properties from P-E loops.