To find the difference, we need to subtract the second fraction from the first fraction.
The first fraction is: $\frac{5x}{x^2 - x - 6}$
The second fraction is: $\frac{4}{x^2 + 4x + 4}$
To subtract these fractions, we need to find a common denominator. The denominators are $(x^2 - x - 6)$ and $(x^2 + 4x + 4)$.
The common denominator is the product of these two denominators: $(x^2 - x - 6)(x^2 + 4x + 4)$.
Now, let's simplify each fraction by multiplying the numerator and denominator by the common denominator.
The first fraction becomes: $\frac{5x(x^2 + 4x + 4)}{(x^2 - x - 6)(x^2 + 4x + 4)}$
The second fraction becomes: $\frac{4(x^2 - x - 6)}{(x^2 - x - 6)(x^2 + 4x + 4)}$
Now, let's subtract these fractions.
$\frac{5x(x^2 + 4x + 4)}{(x^2 - x - 6)(x^2 + 4x + 4)} - \frac{4(x^2 - x - 6)}{(x^2 - x - 6)(x^2 + 4x + 4)}$
To simplify the numerator, we distribute 5x to each term in the parentheses.
$\frac{5x^3 + 20x^2 + 20x}{(x^2 - x - 6)(x^2 + 4x + 4)} - \frac{4(x^2 - x - 6)}{(x^2 - x - 6)(x^2 + 4x + 4)}$
Now, we can combine like terms in the numerator.
$\frac{5x^3 + 20x^2 + 20x - 4x^2 + 4x + 24}{(x^2 - x - 6)(x^2 + 4x + 4)}$
Next, we can combine like terms in the numerator.
$\frac{5x^3 + 16x^2 + 24x + 24}{(x^2 - x - 6)(x^2 + 4x + 4)}$
Finally, we can simplify the fraction by factoring the denominator.
$(x^2 - x - 6)$ can be factored as $(x - 3)(x + 2)$.
$(x^2 + 4x + 4)$ can be factored as $(x + 2)(x + 2)$.
$\frac{5x^3 + 16x^2 + 24x + 24}{(x - 3)(x + 2)(x + 2)}$
The difference is $\boxed{\frac{5x^3 + 16x^2 + 24x + 24}{(x - 3)(x + 2)(x + 2)}}$.
Find the difference.
StartFraction 5x over x squared minus x minus 6 EndFraction minus StartFraction 4 over x squared plus 4 x plus 4 EndFraction
(1 point)
Responses
A. StartFraction 5x minus 4 over left-parenthesis x minus 3 right-parenthesis left-parenthesis x plus 2 right-parenthesis squared EndFraction
Image with alt text: StartFraction 5x minus 4 over left-parenthesis x minus 3 right-parenthesis left-parenthesis x plus 2 right-parenthesis squared EndFraction
B. StartFraction 5 x minus 4 over left-parenthesis x minus 3 right-parenthesis EndFraction
Image with alt text: StartFraction 5 x minus 4 over left-parenthesis x minus 3 right-parenthesis EndFraction
C. StartFraction 5 x squared plus 6 x plus 12 over left-parenthesis x minus 3 right-parenthesis left-parenthesis x plus 2 right-parenthesis EndFraction
Image with alt text: StartFraction 5 x squared plus 6 x plus 12 over left-parenthesis x minus 3 right-parenthesis left-parenthesis x plus 2 right-parenthesis EndFraction
D. StartFraction 5 x squared plus 6 x plus 12 over left-parenthesis x minus 3 right-parenthesis left-parenthesis x plus 2 right-parenthesis squared EndFraction
1 answer