ar quality = ["Старый ", "Новый ", "Долгий "];

var val = document.getElementById("name").innerHTML;

Старый плуг

плуг

var quality = ["Старый ", "Новый ", "Долгий "];
var val = document.getElementById("name").innerHTML;
for each (word in quality) {
    if val.startsWith(word){
        //удаляем слово
        break;
        }
    }