Question

1. \\(\frac{x^2 + 8x + 12}{x^2 + 3x - 18}\\) 16) \\(\frac{x^2 - 11x + 18}{x^2 + 2x - 8}\\) 17) \\(\frac{b^2 + 3b - 28}{b^2 - 49}\\) 18) \\(\frac{v^2 - 3v - 40}{v^2 - 11v + 24}\\) 19) \\(\frac{4n - 4}{6n - 20}\\) 20) \\(\frac{v^2 - 5v - 14}{v^2 + 4v + 4}\\)

Explanation:

Problem 15: $\boldsymbol{\frac{x^2 + 8x + 12}{x^2 + 3x - 18}}$

Step1: Factor numerator

Find two numbers that multiply to $12$ and add to $8$.
$x^2 + 8x + 12 = (x + 2)(x + 6)$

Step2: Factor denominator

Find two numbers that multiply to $-18$ and add to $3$.
$x^2 + 3x - 18 = (x + 6)(x - 3)$

Step3: Simplify the fraction

Cancel the common factor $(x + 6)$ (where $x
eq -6, 3$).
$\frac{(x + 2)(x + 6)}{(x + 6)(x - 3)} = \frac{x + 2}{x - 3}$

Problem 16: $\boldsymbol{\frac{x^2 - 11x + 18}{x^2 + 2x - 8}}$

Step1: Factor numerator

Find two numbers that multiply to $18$ and add to $-11$.
$x^2 - 11x + 18 = (x - 2)(x - 9)$

Step2: Factor denominator

Find two numbers that multiply to $-8$ and add to $2$.
$x^2 + 2x - 8 = (x + 4)(x - 2)$

Step3: Simplify the fraction

Cancel the common factor $(x - 2)$ (where $x
eq 2, -4$).
$\frac{(x - 2)(x - 9)}{(x + 4)(x - 2)} = \frac{x - 9}{x + 4}$

Problem 17: $\boldsymbol{\frac{b^2 + 3b - 28}{b^2 - 49}}$

Step1: Factor numerator

Find two numbers that multiply to $-28$ and add to $3$.
$b^2 + 3b - 28 = (b + 7)(b - 4)$

Step2: Factor denominator (difference of squares)

$b^2 - 49 = (b + 7)(b - 7)$

Step3: Simplify the fraction

Cancel the common factor $(b + 7)$ (where $b
eq \pm 7$).
$\frac{(b + 7)(b - 4)}{(b + 7)(b - 7)} = \frac{b - 4}{b - 7}$

Problem 18: $\boldsymbol{\frac{v^2 - 3v - 40}{v^2 - 11v + 24}}$

Answer:

Step1: Factor numerator

Find two numbers that multiply to $-14$ and add to $-5$.
$v^2 - 5v - 14 = (v - 7)(v + 2)$

Step2: Factor denominator (perfect square)

$v^2 + 4v + 4 = (v + 2)^2$

Step3: Simplify the fraction

Cancel the common factor $(v + 2)$ (where $v
eq -2$).
$\frac{(v - 7)(v + 2)}{(v + 2)^2} = \frac{v - 7}{v + 2}$

Final Answers:

1. $\boldsymbol{\frac{x + 2}{x - 3}}$
2. $\boldsymbol{\frac{x - 9}{x + 4}}$
3. $\boldsymbol{\frac{b - 4}{b - 7}}$
4. $\boldsymbol{\frac{v + 5}{v - 3}}$
5. $\boldsymbol{\frac{2(n - 1)}{3n - 10}}$
6. $\boldsymbol{\frac{v - 7}{v + 2}}$