1. The problem asks to expand and simplify the given algebraic expressions using notable identities. 2. Important identities to use: - Difference of squares: $ (a - b)(a + b) = a^2 - b^2 $ - Square of a binomial: $ (a + b)^2 = a^2 + 2ab + b^2 $ - Square of a binomial: $ (a - b)^2 = a^2 - 2ab + b^2 $ 3. Now, expand each expression step-by-step: - $ (x - 3)(x + 3) = x^2 - 3^2 = x^2 - 9 $ - $ (x + 7)^2 = x^2 + 2 \cdot x \cdot 7 + 7^2 = x^2 + 14x + 49 $ - $ (x - 2)^2 = x^2 - 2 \cdot x \cdot 2 + 2^2 = x^2 - 4x + 4 $ - $ (x + 2)(x - 2) = x^2 - 2^2 = x^2 - 4 $ - $ (x + 4)^2 = x^2 + 2 \cdot x \cdot 4 + 4^2 = x^2 + 8x + 16 $ - $ (x - 7)^2 = x^2 - 2 \cdot x \cdot 7 + 7^2 = x^2 - 14x + 49 $ - $ (2x + 3)^2 = (2x)^2 + 2 \cdot 2x \cdot 3 + 3^2 = 4x^2 + 12x + 9 $ - $ (3x^4 + 3)^2 = (3x^4)^2 + 2 \cdot 3x^4 \cdot 3 + 3^2 = 9x^8 + 18x^4 + 9 $ - $ (x^2 - 5)(x^2 + 5) = (x^2)^2 - 5^2 = x^4 - 25 $ - $ (2x - 3)^2 = (2x)^2 - 2 \cdot 2x \cdot 3 + 3^2 = 4x^2 - 12x + 9 $ - $ (5x + 2)(5x - 2) = (5x)^2 - 2^2 = 25x^2 - 4 $ - $ (-7x + 2)^2 = (-7x)^2 + 2 \cdot (-7x) \cdot 2 + 2^2 = 49x^2 - 28x + 4 $ - $ (-x - 4)^2 = (-x)^2 + 2 \cdot (-x) \cdot (-4) + (-4)^2 = x^2 + 8x + 16 $ - $ (4x^5 - 7)(4x^5 + 7) = (4x^5)^2 - 7^2 = 16x^{10} - 49 $ Final answers: $ (x - 3)(x + 3) = x^2 - 9 $ $ (x + 7)^2 = x^2 + 14x + 49 $ $ (x - 2)^2 = x^2 - 4x + 4 $ $ (x + 2)(x - 2) = x^2 - 4 $ $ (x + 4)^2 = x^2 + 8x + 16 $ $ (x - 7)^2 = x^2 - 14x + 49 $ $ (2x + 3)^2 = 4x^2 + 12x + 9 $ $ (3x^4 + 3)^2 = 9x^8 + 18x^4 + 9 $ $ (x^2 - 5)(x^2 + 5) = x^4 - 25 $ $ (2x - 3)^2 = 4x^2 - 12x + 9 $ $ (5x + 2)(5x - 2) = 25x^2 - 4 $ $ (-7x + 2)^2 = 49x^2 - 28x + 4 $ $ (-x - 4)^2 = x^2 + 8x + 16 $ $ (4x^5 - 7)(4x^5 + 7) = 16x^{10} - 49