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\frac{\left(x^{2}-4x+3\right)\left(x^{2}+x-6\right)}{\left(x^{2}+5x+6\right)\left(x^{2}-5+6\right)}
Divide \frac{x^{2}-4x+3}{x^{2}+5x+6} by \frac{x^{2}-5+6}{x^{2}+x-6} by multiplying \frac{x^{2}-4x+3}{x^{2}+5x+6} by the reciprocal of \frac{x^{2}-5+6}{x^{2}+x-6}.
\frac{\left(x^{2}-4x+3\right)\left(x^{2}+x-6\right)}{\left(x^{2}+5x+6\right)\left(x^{2}+1\right)}
Add -5 and 6 to get 1.
\frac{\left(x-3\right)\left(x-2\right)\left(x-1\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x^{2}+1\right)}
Factor the expressions that are not already factored.
\frac{\left(x-3\right)\left(x-2\right)\left(x-1\right)}{\left(x+2\right)\left(x^{2}+1\right)}
Cancel out x+3 in both numerator and denominator.
\frac{x^{3}-6x^{2}+11x-6}{x^{3}+2x^{2}+x+2}
Expand the expression.
\frac{\left(x^{2}-4x+3\right)\left(x^{2}+x-6\right)}{\left(x^{2}+5x+6\right)\left(x^{2}-5+6\right)}
Divide \frac{x^{2}-4x+3}{x^{2}+5x+6} by \frac{x^{2}-5+6}{x^{2}+x-6} by multiplying \frac{x^{2}-4x+3}{x^{2}+5x+6} by the reciprocal of \frac{x^{2}-5+6}{x^{2}+x-6}.
\frac{\left(x^{2}-4x+3\right)\left(x^{2}+x-6\right)}{\left(x^{2}+5x+6\right)\left(x^{2}+1\right)}
Add -5 and 6 to get 1.
\frac{\left(x-3\right)\left(x-2\right)\left(x-1\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x^{2}+1\right)}
Factor the expressions that are not already factored.
\frac{\left(x-3\right)\left(x-2\right)\left(x-1\right)}{\left(x+2\right)\left(x^{2}+1\right)}
Cancel out x+3 in both numerator and denominator.
\frac{x^{3}-6x^{2}+11x-6}{x^{3}+2x^{2}+x+2}
Expand the expression.