Question

type the correct answer in each box. use numerals instead of words. multiply the expressions. \\(\frac{3x^2 + 2x - 21}{-2x^2 - 2x + 12} cdot \frac{2x^2 + 25x + 63}{6x^2 + 7x - 49}\\) if \\(a = 1\\), find the values of \\(b\\), \\(c\\), and \\(d\\) that make the given expression equivalent to the expression below. \\(\frac{ax + b}{cx + d}\\) \\(b = \square\\), \\(c = \square\\), and \\(d = \square\\)

Explanation:

Step1: Factor first numerator

$3x^2+2x-21=(3x-7)(x+3)$

Step2: Factor first denominator

$-2x^2-2x+12=-2(x^2+x-6)=-2(x+3)(x-2)$

Step3: Factor second numerator

$2x^2+25x+63=(2x+7)(x+9)$

Step4: Factor second denominator

$6x^2+7x-49=(3x-7)(2x+7)$

Step5: Substitute factors into product

$\frac{(3x-7)(x+3)}{-2(x+3)(x-2)} \cdot \frac{(2x+7)(x+9)}{(3x-7)(2x+7)}$

Step6: Cancel common factors

Cancel $(3x-7)$, $(x+3)$, $(2x+7)$:
$\frac{x+9}{-2(x-2)} = \frac{x+9}{-2x+4}$

Step7: Match to $\frac{ax+b}{cx+d}$ (a=1)

$\frac{1x+9}{-2x+4}$

Answer:

$b = 9$, $c = -2$, and $d = 4$