a² + b² = (a + b)² - 2ab = 5² - 2*6 = 25 - 12 = 13.

a³ + b³ = 1³ + 2³ = 1 + 8 = 9.

a³ + b³ = 1³ + 4³ = 1 + 64 = 65.

(a + b)² = (2 + 3)² = 25; a² + b² = 2² + 3² = 4 + 9 = 13; thus, 25 - 13 = 12.

(a + b)² = (2 + 3)² = 5² = 25.

(a + b)² = (2 + 5)² = 7² = 49.

a² + 2ab + b² = 2² + 2*2*5 + 5² = 4 + 20 + 25 = 49.

a² + b² = 3² + 4² = 9 + 16 = 25.

a² + b² - 2ab = 3² + 5² - 2(3)(5) = 9 + 25 - 30 = 4.

a² + b² = 4² + 5² = 16 + 25 = 41.

(a - b)² = (5 - 2)² = 3² = 9.

(a + b)² = (5 + 4)² = 9² = 81.

x² + y² = (x + y)² - 2xy = 10² - 2*21 = 100 - 42 = 58.

x² - y² = 4² - 5² = 16 - 25 = -9.

(x - 1)² = (5 - 1)² = 4² = 16.

(2x - 3)² = 4x² - 12x + 9 by the square of a binomial.

(2x - 3y)² = 4x² - 12xy + 9y² by applying the square of a binomial.

(x + 2)(x - 2) = x² - 2² = x² - 4 by the difference of squares.

(2x + 3)(2x - 3) = 4x² - 9 by the difference of squares.

(a - b)(a + b) = a² - b², which is the difference of squares.

(x + 1)² - (x - 1)² = [(x² + 2x + 1) - (x² - 2x + 1)] = 4x.

(x + y + z)² = x² + y² + z² + 2(xy + xz + yz) is the correct expansion.

(x + y)(x - y) = x² - y², which is the difference of squares.

(2x + 3)(2x - 3) = 4x² - 9.

(2x + 3)² = 4x² + 12x + 9 by expanding the square of a binomial.

(3x + 4)(3x - 4) = (3x)² - 4² = 9x² - 16.

(a + b)² - (a² + b²) = 2ab.

(a + b)² = a² + 2ab + b² by the expansion of the square of a binomial.

(x + 1)((x + 2) + (x - 2)) = (x + 1)(2x) = 2x² + 2x.

(x + 1)(x - 1) = x² - 1, which is the difference of squares.