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ScanSolve AI Tutor · Accepted Answer

To simplify the expression, we will factorize each polynomial, convert the division into multiplication, and then cancel out common factors.

Factorize all quadratic expressions.
Factorize the first numerator, $x^2-1$ :
$x^2-1 = (x-1)(x+1)$
Factorize the first denominator, $x^2+x-2$ :
$x^2+x-2 = (x+2)(x-1)$
Factorize the second numerator, $x^2-4x+3$ :
$x^2-4x+3 = (x-3)(x-1)$
Factorize the second denominator, $x^2+4x+4$ :
$x^2+4x+4 = (x+2)^2 = (x+2)(x+2)$
Substitute the factored expressions back into the original problem.
$\frac{(x-1)(x+1)}{(x+2)(x-1)} \div \frac{(x-3)(x-1)}{(x+2)(x+2)}$
Convert the division into multiplication by inverting the second fraction.
$\frac{(x-1)(x+1)}{(x+2)(x-1)} \times \frac{(x+2)(x+2)}{(x-3)(x-1)}$
Cancel out common factors from the numerator and the denominator.
Cancel one $(x-1)$ from the numerator and one $(x-1)$ from the denominator.
Cancel one $(x+2)$ from the numerator and one $(x+2)$ from the denominator.
$= \frac{\cancel{(x-1)}(x+1)}{\cancel{(x+2)}\cancel{(x-1)}} \times \frac{\cancel{(x+2)}(x+2)}{(x-3)(x-1)}$ $= \frac{(x+1)}{1} \times \frac{(x+2)}{(x-3)(x-1)}$
Multiply the remaining terms to get the simplified expression.
$= \frac{(x+1)(x+2)}{(x-3)(x-1)}$
The simplified expression is:
$\frac{(x+1)(x+2)}{(x-3)(x-1)}$

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