Answer :
To simplify the rational expression [tex]\(\frac{5x^2 - 5}{3x^2 + 3x}\)[/tex], let's go through the steps one by one:
1. Factor the numerator and the denominator:
The numerator is [tex]\(5x^2 - 5\)[/tex]. This can be factored as:
[tex]\[ 5x^2 - 5 = 5(x^2 - 1) \][/tex]
Notice that [tex]\(x^2 - 1\)[/tex] is a difference of squares, which can be factored further:
[tex]\[ 5(x^2 - 1) = 5(x - 1)(x + 1) \][/tex]
The denominator is [tex]\(3x^2 + 3x\)[/tex]. This can be factored by factoring out the common term [tex]\(3x\)[/tex]:
[tex]\[ 3x^2 + 3x = 3x(x + 1) \][/tex]
2. Rewrite the rational expression with these factored forms:
[tex]\[ \frac{5x^2 - 5}{3x^2 + 3x} = \frac{5(x - 1)(x + 1)}{3x(x + 1)} \][/tex]
3. Cancel the common factors in the numerator and the denominator:
Both the numerator and the denominator have a common factor of [tex]\(x + 1\)[/tex]:
[tex]\[ \frac{5(x - 1)\cancel{(x + 1)}}{3x\cancel{(x + 1)}} = \frac{5(x - 1)}{3x} \][/tex]
So, the simplified form of the rational expression [tex]\(\frac{5x^2 - 5}{3x^2 + 3x}\)[/tex] is [tex]\(\frac{5(x - 1)}{3x}\)[/tex].
Thus, the correct answer is:
D. [tex]\(\frac{5(x-1)}{3 x}\)[/tex]
1. Factor the numerator and the denominator:
The numerator is [tex]\(5x^2 - 5\)[/tex]. This can be factored as:
[tex]\[ 5x^2 - 5 = 5(x^2 - 1) \][/tex]
Notice that [tex]\(x^2 - 1\)[/tex] is a difference of squares, which can be factored further:
[tex]\[ 5(x^2 - 1) = 5(x - 1)(x + 1) \][/tex]
The denominator is [tex]\(3x^2 + 3x\)[/tex]. This can be factored by factoring out the common term [tex]\(3x\)[/tex]:
[tex]\[ 3x^2 + 3x = 3x(x + 1) \][/tex]
2. Rewrite the rational expression with these factored forms:
[tex]\[ \frac{5x^2 - 5}{3x^2 + 3x} = \frac{5(x - 1)(x + 1)}{3x(x + 1)} \][/tex]
3. Cancel the common factors in the numerator and the denominator:
Both the numerator and the denominator have a common factor of [tex]\(x + 1\)[/tex]:
[tex]\[ \frac{5(x - 1)\cancel{(x + 1)}}{3x\cancel{(x + 1)}} = \frac{5(x - 1)}{3x} \][/tex]
So, the simplified form of the rational expression [tex]\(\frac{5x^2 - 5}{3x^2 + 3x}\)[/tex] is [tex]\(\frac{5(x - 1)}{3x}\)[/tex].
Thus, the correct answer is:
D. [tex]\(\frac{5(x-1)}{3 x}\)[/tex]
