SUP03-09

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\[A(x) = x^2 + 2x + 1 = (x+1)^2.\] \begin{align*} B(x) &= (2x-5)^2 - x^2& \\ &=\big[(2x-5) - x\big]\big[(2x-5) + x\big]& \\ &=(2x - 5 - x)(2x - 5 + x)& \\ &=(x - 5)(3x - 5).& \end{align*} \begin{align*} C(x) &= 9x^2 + 12x + 4& \\ &=(3x)^2 + 2\cdot (3x) \cdot 2 + 2^2& \\ &=(3x + 2)^2.& \end{align*} \begin{align*} D(x) &= (2x-1)^2 - (x-3)^2& \\ &=\big[(2x - 1) - (x-3)\big]\big[(2x-1) + (x-3)\big]& \\ &=(2x-1-x+3)(2x - 1 + x - 3)& \\ &=(x+2)(3x-4).& \end{align*} \begin{align*} E(x) &= 16(x+1)^2 - 25x^2& \\ &=\big[4(x+1)\big]^2 - (5x)^2& \\ &=\big[4(x+1) - 5x\big]\big[4(x+1) + 5x\big]& \\ &=(4x + 4 - 5x)(4x + 4 + 5x)& \\ &=(-x + 4)(9x+4).& \end{align*} \begin{align*} F(x) &= x^2 - 4 + (x-2)(x+1)& \\ &=\underline{(x-2)}(x+2) + \underline{(x-2)}(x+1)& \\ &=(x-2)\big[(x+2) + (x+1)\big]& \\ &=(x-2)(x+2+x+1)& \\ &=(x-2)(2x + 3).& \end{align*} \begin{align*} G(x) &= 3x^2 - 12x + 12& \\ &=3(x^2 - 4x + 4)& \\ &=3(x^2 - 2\cdot x \cdot 2 + 2^2)& \\ &=3(x-2)^2.& \end{align*} \begin{align*} H(x) &=2x(x+3) + 4x + 12& \\ &=2x\underline{(x+3)} + 4\underline{(x + 3)}& \\ &=(x+3)(2x + 4)& \\ &=2(x+3)(x+2).& \end{align*}

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