To create a rational expression equivalent to the given one [tex]\( \frac{5x^2 + 25x + 20}{7x} \)[/tex], let’s break down the process step by step:
1. Identify the numerator and denominator:
We start by clearly identifying our numerator and denominator from the given rational expression.
- Numerator: [tex]\( 5x^2 + 25x + 20 \)[/tex]
- Denominator: [tex]\( 7x \)[/tex]
2. Consider the list of expressions given to place correctly. We have:
- [tex]\( 7x \)[/tex]
- [tex]\( x + 4 \)[/tex]
- [tex]\( 5(x-1) \)[/tex]
- [tex]\( x-1 \)[/tex]
- [tex]\( 5 x^2 + 15 x - 20 \)[/tex]
3. Analyze and match the numerator and the denominator with the options:
- The denominator in our given expression is [tex]\( 7x \)[/tex]. From the list, [tex]\( 7x \)[/tex] exactly matches our denominator.
- For the numerator, we have [tex]\( 5x^2 + 25x + 20 \)[/tex]. We should try to match or simplify expressions given to see if one matches our numerator.
Analyzing each of the expressions in the numerator list:
- [tex]\( x + 4 \)[/tex]: This does not match our numerator.
- [tex]\( 5(x-1) \)[/tex]: This expands to [tex]\( 5x - 5 \)[/tex], which does not match our numerator.
- [tex]\( x - 1 \)[/tex]: This does not match our numerator.
- [tex]\( 5 x^2 + 15 x - 20 \)[/tex]: This almost matches a style of expression you would expect in quadratic format but doesn't simplify to our exact given numerator.
After this analysis, none of the provided expressions directly come to match our [tex]\( 5x^2 + 25x + 20 \)[/tex]. Indeed, given our numerator [tex]\( 5x^2 + 25x + 20 \)[/tex], we directly use it as it is in our solved answer.
4. Combine the correct numerator and denominator:
- Numerator: [tex]\( 5x^2 + 25x + 20 \)[/tex]
- Denominator: [tex]\( 7x \)[/tex]
Therefore, the correct matching for the given rational expression [tex]\( \frac{5x^2 + 25x + 20}{7x} \)[/tex] considering provided options is only [tex]\( 7x \)[/tex] as the denominator matches directly.
So, the well-organized detailed answer is simply confirming:
- For [tex]\( \frac{5 x^2 + 25 x + 20}{7 x} \)[/tex],
- The correct numerator is [tex]\( 5 x^2 + 25 x + 20 \)[/tex].
- The correct denominator is [tex]\( 7 x \)[/tex].
