[tex]$\frac{50-2 w^{2}}{3 w^{2}+9 w-30} \cdot \frac{w^{2}+5 w-14}{6 w-30}=-\frac{w+7}{9}[/tex]
Solution:
Given expression:
[tex]$\frac{50-2 w^{2}}{3 w^{2}+9 w-30} \cdot \frac{w^{2}+5 w-14}{6 w-30}[/tex]
To solve the given expression:
First simplify: [tex]\frac{50-2 w^{2}}{3 w^{2}+9 w-30}[/tex]
[tex]$\frac{50-2 w^{2}}{3 w^{2}+9 w-30}=-\frac{2(w+5)(w-5)}{3(w-2)(w+5)}[/tex]
Cancel the common factor (w + 5).
[tex]$=-\frac{2(w-5)}{3(w-2)}[/tex]
Now substitute this in the given expression.
[tex]$\frac{50-2 w^{2}}{3 w^{2}+9 w-30} \cdot \frac{w^{2}+5 w-14}{6 w-30}=-\frac{2(w-5)}{3(w-2)} \cdot \frac{w^{2}+5 w-14}{6 w-30}[/tex]
Multiply the fractions [tex]\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d}[/tex]
[tex]$=-\frac{2(w-5)\left(w^{2}+5 w-14\right)}{3(w-2)(6 w-30)}[/tex]
Factor the denominator [tex]3(w-2)(6 w-30) =18(w-2)(w-5)[/tex]
[tex]$=-\frac{2(w-5)\left(w^{2}+5 w-14\right)}{18(w-2)(w-5)}[/tex]
Cancel the common factor 2(w – 5).
[tex]$=-\frac{w^{2}+5 w-14}{9(w-2)}[/tex]
Factor the numerator [tex]w^{2}+5 w-14=(w-2)(w+7)[/tex]
[tex]$=-\frac{(w-2)(w+7)}{9(w-2)}[/tex]
Cancel the common factor (w – 2).
[tex]$=-\frac{w+7}{9}[/tex]
[tex]$\frac{50-2 w^{2}}{3 w^{2}+9 w-30} \cdot \frac{w^{2}+5 w-14}{6 w-30}=-\frac{w+7}{9}[/tex]
