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\(P=\left(\frac{x}{\left(x+2\right)\left(x-2\right)}+\frac{1}{x+2}-\frac{2}{x-2}\right):\left(1-\frac{x}{x+2}\right)\)
\(P=\left(\frac{x}{\left(x+2\right)\left(x-2\right)}+\frac{1\cdot\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}-\frac{2\cdot\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}\right):\left(\frac{1\cdot\left(x+2\right)}{x+2}-\frac{x}{x+2}\right)\)
\(P=\frac{x+x-2-2x-4}{\left(x+2\right)\left(x-2\right)}:\frac{x+2-x}{x+2}\)
\(P=\frac{-6}{\left(x+2\right)\left(x-2\right)}:\frac{2}{x+2}\)
\(P=\frac{-6}{\left(x+2\right)\left(x-2\right)}\cdot\frac{x+2}{2}\)
\(P=\frac{-3}{x-2}.\)
(x + 1)^4 - 6(x + 1)^2 - (x^2 - 2)(x^2 + 2)
= (x^2 + 2x + 1)(x^2 + 2x + 1) - 6(x^2 + 2x + 1) - (x^2 - 2)(x^2 + 2)
= x^2.(x^2 + 2x + 1) + 2x.(x^2 + 2x + 1) + x^2 + 2x + 1 - (x^2 - 2)(x^2 + 2)
= x^4 + 2x^3 + x^2 + 2x^3 + 4x^2 + 2x + x^2 + 2x + 1 - 6x^2 - 12x - 6 - x^2 + 2^2
= 4x^3 - 8x - 1
\(\left(x+1\right)^4-6\left(x+1\right)^2-\left(x^2-2\right)\left(x^2+2\right)\)
\(=\left(x^2+2x-5\right)\left(x^2+2x+1\right)-x^4+2\)
\(=x^4+2x^3+x^2+2x^3+4x^2+2x-5x^2-10x-5-x^4+4\)
\(=4x^3-8x-1\)
\(\left(x-y\right)^2+\left(x+y\right)^2-2\left(x+y\right)\left(x-y\right)-4x^2\)
\(=\left(x-y-x-y\right)^2-\left(2x\right)^2\)
\(=\left(-2y^2\right)-\left(2x\right)^2=\left(2y\right)^2-\left(2x\right)^2=\left(2y-2x\right)\left(2y+2x\right)=4\left(y-x\right)\left(x+y\right)\)
\(\left(x+8\right)^2-2\left(x+8\right)\left(x-2\right)+\left(x-2\right)^2\)
\(=\left(x+8-x+2\right)^2\)
\(=10^2\)
\(=100\)
\(\left(x-2\right)^2-\left(x+2\right)\left(x-2\right)=\left(x^2-4x+4\right)-\left(x^2-4\right)=x^2-4x+4-x^2+4=8-4x\)
(x+2)2-(x+2)(x-2)
=x2+4x+4-x2+4
=4x+8