\( \left. \begin{array} { l } { ( a + 4 b ) ^ { 2 } ( a - 4 b ) ^ { 2 } } \\ { ( a ^ { 2 } + 4 a b + 16 b ^ { 2 } ) ( a ^ { 2 } - 8 a b - 16 b ^ { 2 } ) } \\ { a ^ { 4 } - 64 a b - 256 b ^ { 2 } } \end{array} \right. \)

Simplify the expression by following steps: - step0: : \(a^{4}-64ab-256b^{2}\) Expand the expression \( (a + 4b)^2 (a - 4b)^2 \) Simplify the expression by following steps: - step0: Calculate: \(\left(a+4b\right)^{2}\left(a-4b\right)^{2}\) - step1: Multiply the expression: \(\left(a^{2}-16b^{2}\right)^{2}\) - step2: Expand the expression: \(\left(a^{2}\right)^{2}-2a^{2}\times 16b^{2}+\left(16b^{2}\right)^{2}\) - step3: Calculate: \(a^{4}-32a^{2}b^{2}+256b^{4}\) Expand the expression \( (a^2 + 4ab + 16b^2)(a^2 - 8ab - 16b^2) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(a^{2}+4ab+16b^{2}\right)\left(a^{2}-8ab-16b^{2}\right)\) - step1: Apply the distributive property: \(a^{2}\times a^{2}-a^{2}\times 8ab-a^{2}\times 16b^{2}+4aba^{2}-4ab\times 8ab-4ab\times 16b^{2}+16b^{2}a^{2}-16b^{2}\times 8ab-16b^{2}\times 16b^{2}\) - step2: Multiply the terms: \(a^{4}-8a^{3}b-16a^{2}b^{2}+4a^{3}b-32a^{2}b^{2}-64ab^{3}+16b^{2}a^{2}-128b^{3}a-256b^{4}\) - step3: Add the terms: \(a^{4}-4a^{3}b-32a^{2}b^{2}-192ab^{3}-256b^{4}\) Let's break down the expressions step by step: 1. **First Expression: \( (a + 4b)^2 (a - 4b)^2 \)** We can expand this expression using the formula for the difference of squares: \[ (x+y)(x-y) = x^2 - y^2 \] Here, let \( x = (a + 4b) \) and \( y = (a - 4b) \). Thus, we have: \[ (a + 4b)^2 (a - 4b)^2 = ((a + 4b)(a - 4b))^2 = (a^2 - (4b)^2)^2 = (a^2 - 16b^2)^2 \] Now, expanding \( (a^2 - 16b^2)^2 \): \[ = a^4 - 32a^2b^2 + 256b^4 \] 2. **Second Expression: \( (a^2 + 4ab + 16b^2)(a^2 - 8ab - 16b^2) \)** We will expand this expression using the distributive property: \[ = a^2(a^2 - 8ab - 16b^2) + 4ab(a^2 - 8ab - 16b^2) + 16b^2(a^2 - 8ab - 16b^2) \] Expanding each term: - \( a^2(a^2 - 8ab - 16b^2) = a^4 - 8a^3b - 16a^2b^2 \) - \( 4ab(a^2 - 8ab - 16b^2) = 4a^3b - 32a^2b^2 - 64ab^3 \) - \( 16b^2(a^2 - 8ab - 16b^2) = 16a^2b^2 - 128ab^3 - 256b^4 \) Combining all these terms: \[ = a^4 - 4a^3b - 32a^2b^2 - 192ab^3 - 256b^4 \] 3. **Third Expression: \( a^4 - 64ab - 256b^2 \)** This expression is already in its simplest form: \[ = a^4 - 64ab - 256b^2 \] ### Summary of Results: 1. \( (a + 4b)^2 (a - 4b)^2 = a^4 - 32a^2b^2 + 256b^4 \) 2. \( (a^2 + 4ab + 16b^2)(a^2 - 8ab - 16b^2) = a^4 - 4a^3b - 32a^2b^2 - 192ab^3 - 256b^4 \) 3. \( a^4 - 64ab - 256b^2 \) (already simplified)