Show that:
(i) (3x+7)² - 84x = (3x-7)²
(ii) (9p-5q)² + 180pq = (9p+5q)²
(iii) (4/3m - 3/4n)² + 2mn = 169m²/9 + 9/16n²
(iv) (4pq+3q)² - (4pq-3q)² = 48pq²
(v) (a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a) = 0

We know that (a±b)² = a² + b² ± 2ab and (a-b)(a+b) = a² - b²
Now,
(i) (3x+7)² - 84x = (3x)² + 7² + 2(3x)(7) - 84x = (3x)² + 7² - 42x = (3x-7)²
(ii) (9p-5q)² + 180pq = (9p)² + (5q)² - 2(9p)(5q) + 180pq = (9p)² + (5q)² - 90pq + 180pq = (9p)² + (5q)² + 90pq = (9p+5q)²
(iii) (4/3m - 3/4n)² + 2mn = (4/3m)² + (3/4n)² - 2(4/3m)(3/4n) + 2mn = (4/3m)² + (3/4n)² - 2mn + 2mn = 169m²/9 + 9/16n²
(iv) (4pq+3q)² - (4pq-3q)² = (4pq)² + (3q)² + 2(4pq)(3q) - ((4pq)² + (3q)² - 2(4pq)(3q)) = (4pq)² + (3q)² + 2(4pq)(3q) - (4pq)² - (3q)² + 2(4pq)(3q) = 24pq² + 24pq² = 48pq²
(v) (a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a) = a² - b² + b² - c² + c² - a² = 0