Sentence improvement (reported process: backshift to past perfect in a “how”-clause) Original sentence: "We demonstrated to them how we ___ the artistic patterns." Choose the option that yields clear past sequence and grammatical form.

Timeline

• Main action: “We demonstrated …” (simple past).
• Earlier action: preparation of artistic patterns occurred before that demonstration.

Concept/Approach
To show an action completed prior to a past reference point, use the past perfect: had prepared.

Option analysis
had prepared — correct backshift and aspect. ✅are prepared / are preparing — present (simple/progressive); wrong time.have prepared — present perfect; wrong reference time.

Final Answer
“… how we had prepared the artistic patterns.”