To simplify the expression \( \frac{5x^2 + 10x^2 + 7x + 12}{x^2 - 4x^2 + x - 6} \), we first need to correct the denominator because the expression given seems off.
Let's clarify the expression and simplify each part:
Step 1: Combine Like Terms
The numerator is: \[ 5x^2 + 10x^2 + 7x + 12 = 15x^2 + 7x + 12 \]
The denominator can be simplified. Assuming the expression was intended as \( x^2 - 4x + x - 6 \) and combining the like terms yields: \[ (x^2 + x - 4x - 6) = (x^2 - 3x - 6) \]
So now our expression is: \[ \frac{15x^2 + 7x + 12}{x^2 - 3x - 6} \]
Step 2: Factor the Numerator and Denominator
To factor the numerator \( 15x^2 + 7x + 12 \):
We look for two numbers that multiply to \( 15 \times 12 = 180 \) and add to \( 7 \). The pair \( 15 \) and \( 12 \) doesn't seem to help easily.
Let's just use the quadratic formula to find its factors. The roots will be: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 15 \cdot 12}}{2 \cdot 15} \] Calculating this gives complex factors, so we will leave it as is.
To factor the denominator \( x^2 - 3x - 6 \):
We can look for factors. This factors into: \((x - 6)(x + 1)\).
Final Expression:
Since we couldn't factor the numerator neatly, we are left with: \[ \frac{15x^2 + 7x + 12}{(x - 6)(x + 1)} \]
Unfortunately, we cannot simplify further without specific values for x.
Therefore among the responses given and based on workings, the correct answer from your options would be:
- None of the provided responses seem accurate regarding the function calculated above.
To summarize: Check the expressions given for correctness before choosing based on factors.