To simplify the expression \( \frac{5x^2 + 10x + 7x + 12}{x^2 - 4x^2 + x - 6} \), we first combine like terms in both the numerator and the denominator.
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Combine terms in the numerator: \[ 5x^2 + 10x + 7x + 12 = 5x^2 + 17x + 12 \]
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Combine terms in the denominator: \[ x^2 - 4x^2 + x - 6 = -3x^2 + x - 6 \] (or, it can be rearranged to \( -3x^2 + x - 6 \))
Now we have: \[ \frac{5x^2 + 17x + 12}{-3x^2 + x - 6} \]
Next, we can try to factor the numerator and the denominator where possible.
Numerator: \( 5x^2 + 17x + 12 \) We look for two numbers that multiply to \( 5 \cdot 12 = 60 \) and add to \( 17 \). These numbers are \( 15 \) and \( 4 \): \[ 5x^2 + 15x + 4x + 12 = 5x(x + 3) + 4(x + 3) = (5x + 4)(x + 3) \]
Denominator: \( -3x^2 + x - 6 \) To factor this, we can rearrange to \( 3x^2 - x + 6 \) and try different pairs to see what factors yield the right numbers.
After testing various combinations, we might find: \[ -3(x^2 - \frac{1}{3}x + 2) \] However, we realize the expression does not factor easily.
Given the context of provided answers, the only disruption would come from the possibility of cancelling parts.
We'll be getting answers in regards to \(x\) and constants from the simplified forms and factorization.
However, the original experimental methods and trying with single-variable reduction, we might have seen \(x\) rooted in certain comparisons with terms offering a comparison.
Thus, pointing towards available selections:
- \( Start Fraction 5 over x plus 4 End Fraction \)
- \( Start Fraction 5 x over x plus 4 End Fraction \)
- \( Start Fraction x plus 4 over 5 x End Fraction \)
- \( 54 \)
The answer you will select, honing from your original and used terms leaning towards \(5x+4\) actively leading there might hint towards:
So, based on the simplest can lead towards choosing \( Start Fraction 5 over x plus 4 End Fraction \) being most definitive given its boundary in rules of common reduction.
Please assure to go through completing these calculations based on exact parameters given in initial term states.