Question

simplify the following expression: \\(\dfrac{6x^3 - 6x}{x^3 + 5x^2} \div \dfrac{4x^2 + 24x - 28}{10x^4 + 70x^3}\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{6x^3 - 6x}{x^3 + 5x^2} \times \frac{10x^4 + 70x^3}{4x^2 + 24x - 28}$

Step2: Factor all polynomials

• Numerator 1: $6x(x^2 - 1) = 6x(x-1)(x+1)$
• Denominator 1: $x^2(x + 5)$
• Numerator 2: $10x^3(x + 7)$
• Denominator 2: $4(x^2 + 6x - 7) = 4(x+7)(x-1)$

Substitute factored forms:
$\frac{6x(x-1)(x+1)}{x^2(x + 5)} \times \frac{10x^3(x + 7)}{4(x+7)(x-1)}$

Step3: Cancel common factors

Cancel $x$, $(x-1)$, $(x+7)$ across numerators/denominators:
$\frac{6(x+1)}{x(x + 5)} \times \frac{10x^3}{4}$

Step4: Multiply remaining terms

Multiply numerators: $6(x+1) \times 10x^3 = 60x^3(x+1)$
Multiply denominators: $x(x + 5) \times 4 = 4x(x+5)$
Simplify the fraction: $\frac{60x^3(x+1)}{4x(x+5)} = \frac{15x^2(x+1)}{x+5}$

Step5: Expand numerator (optional, simplified form)

$15x^2(x+1) = 15x^3 + 15x^2$
$\frac{15x^3 + 15x^2}{x+5}$

Answer:

$\frac{15x^2(x+1)}{x+5}$ or $\frac{15x^3 + 15x^2}{x+5}$ (where $x
eq -7, -5, 0, 1$)