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fill in the blank x²+2x+1/BLANK x BLANK/7x²+7x

To solve this problem, we need to find the appropriate expressions to create a rational expression equivalent to the one given: \(\frac{5x^2 + 25x + 20}{7x}\).

First, let's simplify the given expression:

\[ \frac{5x^2 + 25x + 20}{7x} \]

Factor the numerator:

\[ 5x^2 + 25x + 20 = 5(x^2 + 5x + 4) \]

Further factor the quadratic expression:

\[ x^2 + 5x + 4 = (x + 1)(x + 4) \]

So, the expression becomes:

\[ \frac{5(x + 1)(x + 4)}{7x} \]

Now, match this with the expressions given:

### For the numerator: \( x^2 + 2x + 1 \)
We need to match \( 5(x + 1)(x + 4) \) with the form \( 5x^2 + 10x + 5 \). But here we notice that \( x^2 + 2x + 1 \) corresponds to \( (x + 1)^2 \).

### For the denominator: \( 7x^2 + 7x \)
Here, the factor \( 7x(x + 1) \) corresponds to \( 7x \).

Hence, the correct pieces to fill in are:

1. \( 5(x + 1) \) should match \( x^2 + 2x + 1 \) by its value.
2. \( x \) should be the second part of the numerator in the blank space.
3. \( 7x(x + 1) \) matches the denominator \( 7x^2 + 7x \).

So, the filled expression would be:

\[ \frac{5(x + 1)(x + 4)}{7x} \]

Thus, the correct blank to fill are:

\[
\frac{(x + 1)(x + 4)}{7x}
\]

We can summarize as follows:
1. Fill in the first blank numerator with \( 5(x + 4) \) matching the second term \( x + 4 \).
2. The denominator blank is \( 7x \).

Finally, the complete filled expression should look like this:

\[
\frac{x + 4}{7x}
\]

Would you like details or have any questions?

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### Questions:
1. How do you factorize a quadratic expression?
2. What steps are involved in simplifying rational expressions?
3. How can you match parts of a rational expression to given terms?
4. What is the importance of factoring in simplifying rational expressions?
5. How do you determine the least common denominator in rational expressions?
6. What is the role of common factors in simplifying expressions?
7. Can you explain the steps to verify a simplified rational expression?
8. How do you handle complex fractions involving multiple terms?

### Tip:
Always factorize both the numerator and the denominator separately before attempting to simplify the rational expression.

Learn how to simplify a rational expression involving a quadratic numerator and linear denominator. This problem requires factoring and matching components of rational expressions. Suitable for Grades 9-12.

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