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Solving a Quadratic Equation by Completion of Squares Method

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Question

Simplify $\frac{(x^{2} + 1) (x^{2} + 2)}{(x^{2} + 3) (x^{2} + 4)} =$

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Solution

$\frac{(x^{2} + 1) (x^{2} + 2)}{(x^{2} + 3) (x^{2} + 4)}$

$= \frac{x^{2} x^{2} + x^{2} \cdot 2 + 1 \cdot x^{2} + 1 \cdot 2}{x^{2} x^{2} + x^{2} \cdot 4 + 3 x^{2} + 3 \cdot 4}$

$= \frac{x^{2} x^{2} + 2 x^{2} + 1 \cdot x^{2} + 1 \cdot 2}{x^{2} x^{2} + 4 x^{2} + 3 x^{2} + 3 \cdot 4}$

$= \frac{x^{4} + 3 x^{2} + 2}{x^{4} + 7 x^{2} + 12}$

$= 1 + \frac{- 4 x^{2} - 10}{x^{4} + 7 x^{2} + 12}$

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