(i) \((x+2)^2\)

x² + 2 (x) (2) + 2²

= x² + 4x + 4

(ii) \((8x+3b)^2\)

(8x)² + 2 (8x) (3b) + (3b)²

= 16x² + 48xb + 9b²

(iii) \((2m+1)^2\)

(2m)² + 2 (2m) (1) + 1²

= 4m² + 4m + 1

(iv) \((9a+\frac{1}{6})^2\)

(9a)² + 2 (9a) (\(\frac{1}{6}\)) + (\(\frac{1}{6}\))²

= 81a² + 3a + \(\frac{1}{36}\)

(v) \((x+\frac{x^2}{2})^2\)

(x)² + 2 (x) (\(\frac{x\times x}{2}\)) + (\(\frac{x\times x}{2}\))²

= x² + x³ + \(\frac{1}{4}\)x⁴

(vi) \((\frac{x}{4}-\frac{y}{3})\)

(\(\frac{x}{4}\))² – 2 (\(\frac{x}{4}\)) (\(\frac{y}{3}\)) + (\(\frac{y}{3}\))²

= \(\frac{1}{16}\)x² - \(\frac{xy}{6}\) + \(\frac{1}{9}\)y²

(vii) \((3x-\frac{1}{3x})^2\)

(3x)² – 2 (3x) (\(\frac{1}{3x}\)) + (\(\frac{1}{3x}\))²

= 9x² – 2 + \(\frac{1}{9\times x\times x}\)

(viii) \((\frac{x}{y}-\frac{y}{x})^2\)

(\(\frac{x}{y}\))² – 2 (\(\frac{x}{y}\)) (\(\frac{y}{x}\)) + (\(\frac{y}{x}\))²

= \(\frac{x\times x}{y\times y}\) - 2 + \(\frac{y\times y}{x\times x}\)

(ix) \((\frac{3a}{2}-\frac{5b}{4})^2\)

(\(\frac{3a}{2}\))² – 2 (\(\frac{3a}{2}\)) (\(\frac{5b}{4}\)) + (\(\frac{5b}{4}\))²

= \(\frac{9}{4}\)a² - \(\frac{15}{4}\)ab + \(\frac{25}{16}\)b

(x) \((a^2b-bc^2)^2\)

(a²b)² – 2 (a²b) (bc²) + (bc²)²

= a⁴b² – 2a²b²c² + b²c⁴

(xi) \((\frac{2a}{3b}+\frac{2b}{3a})^2\)

(\(\frac{2a}{3b}\))² + 2 (\(\frac{2a}{3b}\)) (\(\frac{2b}{3a}\)) + (\(\frac{2b}{3a}\))²

= \(\frac{4\times a\times a}{9\times b\times b}\) + \(\frac{8}{9}\)a + \(\frac{4\times b\times b}{9\times a\times a}\)

(xii) \((x^2-ay)^2\)

(x²)² – 2 (x²) (ay) + (ay)²

= x⁴ – 2x²ay + a²y²