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Rational Expression

Simplify: x...

Question

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

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Solution

$\frac{x^{2} + 7 x + 12}{x^{2} + 4 x + 3} = \frac{x^{2} + 4 x + 3 + 3 x + 9}{x^{2} + 4 x + 3}$

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

We can write $x^{2} + 4 x + 3 = (x + 3) (x + 1)$

$= 1 + \frac{3 (x + 3)}{(x + 1) (x + 3)}$

$= 1 + \frac{3}{(x + 1)}$

$= \frac{x + 1 + 3}{(x + 1)}$
$∴ \frac{x^{2} + 7 x + 12}{x^{2} + 4 x + 3} = \frac{x + 4}{(x + 1)}$

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