1. **Stating the problem:** We are asked to express the given algebraic expressions using inverse operations or to expand and simplify algebraic expressions using appropriate properties. 2. **Using inverse operations for identities:** - For expressions like $x^2 - 25$, recognize it as a difference of squares: $$x^2 - 25 = (x - 5)(x + 5)$$ - Similarly, for $x^2 - 9$: $$x^2 - 9 = (x - 3)(x + 3)$$ - For perfect square trinomials like $x^2 + 8x + 16$: $$x^2 + 8x + 16 = (x + 4)^2$$ - For $4x^2 - 16$: $$4x^2 - 16 = (2x)^2 - 4^2 = (2x - 4)(2x + 4)$$ - For $x^2 + 6x + 9$: $$x^2 + 6x + 9 = (x + 3)^2$$ - For $x^2 - 4x + 4$: $$x^2 - 4x + 4 = (x - 2)^2$$ - For $25x^2 - 16$: $$25x^2 - 16 = (5x)^2 - 4^2 = (5x - 4)(5x + 4)$$ - For $x^2 - 20x + 100$: $$x^2 - 20x + 100 = (x - 10)^2$$ 3. **Expanding algebraic expressions:** - $x^2 \cdot (5 - 3x) = 5x^2 - 3x^3$ - $2x \cdot 4x^3 = 8x^4$ - $-(3x + 3x^2) - x^2 + 7x + 3x^2 = -3x - 3x^2 - x^2 + 7x + 3x^2 = (-3x + 7x) + (-3x^2 - x^2 + 3x^2) = 4x - x^2$ - $(5x - 4) \cdot (-5) = -25x + 20$ - $2x \cdot (x + 4) = 2x^2 + 8x$ - $4x \cdot (3 - x) = 12x - 4x^2$ - $(x + 2)^2 = x^2 + 4x + 4$ - $(x + 4)(x - 4) = x^2 - 16$ - $(x - 5)^2 = x^2 - 10x + 25$ - $2x^2 \cdot 2x^4 \cdot (-2) = 2 \cdot 2 \cdot (-2) x^{2+4} = -8x^6$ - $2x - 3x + 2x^2 = (-x) + 2x^2 = 2x^2 - x$ - $(4x - 2)^2 = (4x)^2 - 2 \cdot 4x \cdot 2 + 2^2 = 16x^2 - 16x + 4$ - $(x - 7)(x + 7) = x^2 - 49$ **Final answers:** - $x^2 - 25 = (x - 5)(x + 5)$ - $x^2 - 9 = (x - 3)(x + 3)$ - $x^2 + 8x + 16 = (x + 4)^2$ - $4x^2 - 16 = (2x - 4)(2x + 4)$ - $x^2 + 6x + 9 = (x + 3)^2$ - $x^2 - 4x + 4 = (x - 2)^2$ - $25x^2 - 16 = (5x - 4)(5x + 4)$ - $x^2 - 20x + 100 = (x - 10)^2$ - $x^2 \cdot (5 - 3x) = 5x^2 - 3x^3$ - $2x \cdot 4x^3 = 8x^4$ - $-(3x + 3x^2) - x^2 + 7x + 3x^2 = 4x - x^2$ - $(5x - 4) \cdot (-5) = -25x + 20$ - $2x \cdot (x + 4) = 2x^2 + 8x$ - $4x \cdot (3 - x) = 12x - 4x^2$ - $(x + 2)^2 = x^2 + 4x + 4$ - $(x + 4)(x - 4) = x^2 - 16$ - $(x - 5)^2 = x^2 - 10x + 25$ - $2x^2 \cdot 2x^4 \cdot (-2) = -8x^6$ - $2x - 3x + 2x^2 = 2x^2 - x$ - $(4x - 2)^2 = 16x^2 - 16x + 4$ - $(x - 7)(x + 7) = x^2 - 49$