What should be added to \(\dfrac{x^2-1}{x^2+1}\) to get \(\dfrac{2x^3+3x^2-1}{(x+2)(x^2+1)}\) ?
\(\dfrac{x+1}{x+2}\)
Let A be the quantity added, so \(\dfrac{x^2-1}{x^2+1}+A=\dfrac{2x^3+3x^2-1}{(x+2)(x^2+1)}\). Thus \(A=\dfrac{2x^3+3x^2-1}{(x+2)(x^2+1)}-\dfrac{x^2-1}{x^2+1}=\dfrac{(2x^3+3x^2-1)-(x^2-1)(x+2)}{(x+2)(x^2+1)}\). Since \((x^2-1)(x+2)=x^3+2x^2-x-2\), the numerator simplifies to \(x^3+x^2+x+1=(x+1)(x^2+1)\). So \(A=\dfrac{(x+1)(x^2+1)}{(x+2)(x^2+1)}=\dfrac{x+1}{x+2}\). The correct option is (c).
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