Question
Express in simplest form: 5x^2+10x/x^2+7x+1^2÷x^2−4/x2^+x−6
(1 point)
Responses
5x +4
5x +4
x+45x
the fraction with numerator x plus 4 and denominator 5 x
54
5 fourths
5xx+4
the fraction with numerator 5 x and denominator x plus 4
(1 point)
Responses
5x +4
5x +4
x+45x
the fraction with numerator x plus 4 and denominator 5 x
54
5 fourths
5xx+4
the fraction with numerator 5 x and denominator x plus 4
Answers
GPT-4o mini
To express the given expression \( \frac{5x^2 + 10x}{x^2 + 7x + 1^2} \div \frac{x^2 - 4}{x^2 + x - 6} \) in simplest form, let's analyze it step by step.
1. **First Part: Simplifying the Numerator and Denominator**:
- The numerator is:
\[ 5x^2 + 10x = 5x(x + 2) \]
- The denominator is:
\[ x^2 + 7x + 1^2 = x^2 + 7x + 1 = x^2 + 7x + 1 \]
This doesn't factor nicely and will remain as is.
- The second part of the expression (the divisor) is:
\[ \frac{x^2 - 4}{x^2 + x - 6} \]
Factoring these:
- \( x^2 - 4 = (x - 2)(x + 2) \) (difference of squares)
- \( x^2 + x - 6 \) can be factored as \( (x + 3)(x - 2) \).
2. **Putting it all together**:
The overall expression can be rewritten as:
\[
\frac{5x(x + 2)}{x^2 + 7x + 1} \div \frac{(x - 2)(x + 2)}{(x + 3)(x - 2)}
\]
Which is the same as multiplying by the reciprocal:
\[
\frac{5x(x + 2)}{x^2 + 7x + 1} \times \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)}
\]
3. **Canceling terms**:
The \( (x + 2) \) cancels with itself, and \( (x - 2) \) in the numerator and the denominator cancels:
\[
\frac{5x \cdot (x + 3)}{x^2 + 7x + 1}
\]
So we have:
\[
\frac{5x(x + 3)}{x^2 + 7x + 1}
\]
4. **Final Check**:
The expression \( x^2 + 7x + 1 \) does not factor easily and can remain as it is.
5. **Conclusion**:
The simplified form, in all likelihood based on your options, doesn't match. However, the answer we derive is:
\[
\frac{5x(x + 3)}{x^2 + 7x + 1}
\]
If you're looking for the matching choice from your options, it appears none of them are represented correctly based on this simplified form.
Please double-check with the provided choices, but based on the simplifications, the final answer doesn't directly correspond to any of those listed.
1. **First Part: Simplifying the Numerator and Denominator**:
- The numerator is:
\[ 5x^2 + 10x = 5x(x + 2) \]
- The denominator is:
\[ x^2 + 7x + 1^2 = x^2 + 7x + 1 = x^2 + 7x + 1 \]
This doesn't factor nicely and will remain as is.
- The second part of the expression (the divisor) is:
\[ \frac{x^2 - 4}{x^2 + x - 6} \]
Factoring these:
- \( x^2 - 4 = (x - 2)(x + 2) \) (difference of squares)
- \( x^2 + x - 6 \) can be factored as \( (x + 3)(x - 2) \).
2. **Putting it all together**:
The overall expression can be rewritten as:
\[
\frac{5x(x + 2)}{x^2 + 7x + 1} \div \frac{(x - 2)(x + 2)}{(x + 3)(x - 2)}
\]
Which is the same as multiplying by the reciprocal:
\[
\frac{5x(x + 2)}{x^2 + 7x + 1} \times \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)}
\]
3. **Canceling terms**:
The \( (x + 2) \) cancels with itself, and \( (x - 2) \) in the numerator and the denominator cancels:
\[
\frac{5x \cdot (x + 3)}{x^2 + 7x + 1}
\]
So we have:
\[
\frac{5x(x + 3)}{x^2 + 7x + 1}
\]
4. **Final Check**:
The expression \( x^2 + 7x + 1 \) does not factor easily and can remain as it is.
5. **Conclusion**:
The simplified form, in all likelihood based on your options, doesn't match. However, the answer we derive is:
\[
\frac{5x(x + 3)}{x^2 + 7x + 1}
\]
If you're looking for the matching choice from your options, it appears none of them are represented correctly based on this simplified form.
Please double-check with the provided choices, but based on the simplifications, the final answer doesn't directly correspond to any of those listed.