Question

perform the following operation and express in simplest form. $\frac{3x}{x - 5} cdot \frac{x^{2} + 4x - 45}{x^{2} + 2x - 63}$

Explanation:

Step1: Factor quadratic numerator

Factor $x^2 + 4x - 45$ into binomials.
$x^2 + 4x - 45 = (x + 9)(x - 5)$

Step2: Factor quadratic denominator

Factor $x^2 + 2x - 63$ into binomials.
$x^2 + 2x - 63 = (x + 9)(x - 7)$

Step3: Substitute factored forms

Replace quadratics with their factored equivalents in the original expression.
$\frac{3x}{x - 5} \cdot \frac{(x + 9)(x - 5)}{(x + 9)(x - 7)}$

Step4: Cancel common factors

Eliminate identical non-zero factors from numerator and denominator.
$\frac{3x}{\cancel{x - 5}} \cdot \frac{\cancel{(x + 9)}\cancel{(x - 5)}}{\cancel{(x + 9)}(x - 7)} = \frac{3x}{x - 7}$

Answer:

$\frac{3x}{x - 7}$ (where $x
eq 5, -9, 7$)