Question
Express in simplest form: 5x2+10xx2+7x+12÷x2−4x2+x−6
.(1 point)
Responses
54
Start Fraction 5 over 4 End Fraction
5x+4
Start Fraction 5 over x plus 4 End Fraction
x+45x
Start Fraction x plus 4 over 5 x End Fraction
5xx+4
.(1 point)
Responses
54
Start Fraction 5 over 4 End Fraction
5x+4
Start Fraction 5 over x plus 4 End Fraction
x+45x
Start Fraction x plus 4 over 5 x End Fraction
5xx+4
Answers
GPT-4o mini
To simplify the expression \( \frac{5x^2 + 10x + 7x + 12}{x^2 - 4x^2 + x - 6} \), we need to start by simplifying both the numerator and the denominator.
### Step 1: Simplify the numerator
The numerator is:
\[ 5x^2 + 10x + 7x + 12 = 5x^2 + 17x + 12 \]
### Step 2: Simplify the denominator
The denominator is:
\[ x^2 - 4x^2 + x - 6 = -3x^2 + x - 6 \]
We can factor out -1:
\[ -(3x^2 - x + 6) \]
### Step 3: Combine the simplified numerator and denominator
Now we have:
\[ \frac{5x^2 + 17x + 12}{-(3x^2 - x + 6)} \]
We can write this as:
\[ -\frac{5x^2 + 17x + 12}{3x^2 - x + 6} \]
### Step 4: Factor the numerator and denominator if possible
**Numerator factoring:**
Looking for two numbers that multiply to \( 5 \cdot 12 = 60 \) and add to \( 17 \):
These numbers are \( 15 \) and \( 4\).
Thus, we can rewrite:
\[ 5x^2 + 15x + 4x + 12 = (5x^2 + 15x) + (4x + 12) = 5x(x + 3) + 4(x + 3) = (5x + 4)(x + 3) \]
**Denominator factoring:**
For \( 3x^2 - x + 6 \), there are no obvious integer factors that work, so we will not factor it for this expression.
### Step 5: Final expression
Putting it all together:
\[ - \frac{(5x + 4)(x + 3)}{(3x^2 - x + 6)} \]
This is already in a simplified form unless \( 3x^2 - x + 6 \) can be factored, which it cannot.
The expression simplifies to:
\[ -\frac{(5x + 4)(x + 3)}{3x^2 - x + 6} \]
Since the answer choices provided seem not to match this derivation, the original question could possibly have some mistakes or alternatives to clarify. Please check the selections provided or let me know for further assistance!
### Step 1: Simplify the numerator
The numerator is:
\[ 5x^2 + 10x + 7x + 12 = 5x^2 + 17x + 12 \]
### Step 2: Simplify the denominator
The denominator is:
\[ x^2 - 4x^2 + x - 6 = -3x^2 + x - 6 \]
We can factor out -1:
\[ -(3x^2 - x + 6) \]
### Step 3: Combine the simplified numerator and denominator
Now we have:
\[ \frac{5x^2 + 17x + 12}{-(3x^2 - x + 6)} \]
We can write this as:
\[ -\frac{5x^2 + 17x + 12}{3x^2 - x + 6} \]
### Step 4: Factor the numerator and denominator if possible
**Numerator factoring:**
Looking for two numbers that multiply to \( 5 \cdot 12 = 60 \) and add to \( 17 \):
These numbers are \( 15 \) and \( 4\).
Thus, we can rewrite:
\[ 5x^2 + 15x + 4x + 12 = (5x^2 + 15x) + (4x + 12) = 5x(x + 3) + 4(x + 3) = (5x + 4)(x + 3) \]
**Denominator factoring:**
For \( 3x^2 - x + 6 \), there are no obvious integer factors that work, so we will not factor it for this expression.
### Step 5: Final expression
Putting it all together:
\[ - \frac{(5x + 4)(x + 3)}{(3x^2 - x + 6)} \]
This is already in a simplified form unless \( 3x^2 - x + 6 \) can be factored, which it cannot.
The expression simplifies to:
\[ -\frac{(5x + 4)(x + 3)}{3x^2 - x + 6} \]
Since the answer choices provided seem not to match this derivation, the original question could possibly have some mistakes or alternatives to clarify. Please check the selections provided or let me know for further assistance!