To simplify the expression \( \frac{5x^2 + 10x^2 + 7x + 12}{x^2 - 4x^2 + x - 6} \), we first need to simplify both the numerator and the denominator.
Numerator: Combine like terms:
\[ 5x^2 + 10x^2 = 15x^2 \]
So the numerator simplifies to:
\[ 15x^2 + 7x + 12 \]
Denominator: Combine like terms:
\[ x^2 - 4x^2 = -3x^2 \]
So the denominator is:
\[ -3x^2 + x - 6 \]
Now the expression becomes:
\[ \frac{15x^2 + 7x + 12}{-3x^2 + x - 6} \]
Next, we'll factor both the numerator and the denominator if possible.
Factoring the numerator:
To factor \( 15x^2 + 7x + 12 \), we can look for factors of \( 15 \times 12 = 180 \) that add up to \( 7 \):
The relevant factors are \( 15 \) and \( 12 \).
Using the method of trial and error or the quadratic formula, we find the factors:
\[ 15x^2 + 7x + 12 = (3x + 4)(5x + 3) \]
Factoring the denominator:
To factor \( -3x^2 + x - 6 \), we can try rearranging it to:
\[ -3x^2 + x - 6 = -1(3x^2 - x + 6) \]
The denominator does not factor nicely in integers, so we have it as \( -1(3x^2 - x + 6) \). It does not yield any simple rational or integer roots.
Finally, rewriting the expression:
\[ \frac{15x^2 + 7x + 12}{-3x^2 + x - 6} = \frac{(3x + 4)(5x + 3)}{-(3x^2 - x + 6)} \]
Given the options, we need to consider which one simplifies correctly and is coherent.
- The numerator simplifies to \( \frac{15x^2 + 7x + 12}{-3x^2 + x - 6} \).
- Check which of the provided choices are equivalent.
The numerator and denominator do not allow simplifications or easy cancellations.
Looking at the options you provided:
- \(5x + 4\)
- \( \frac{5}{x + 4} \)
- \( \frac{5x}{x + 4} \)
- \( x + 4 * 5x \)
After checking all options and since none of the options are equal to the actual rational function we simplified, it's okay to conclude none match exactly as discussed.
The overall expression can be said to have been simplified down to a non-directly matchable structure but aligns with the expression \(\frac{3x + 4}{-(3x^2 - x + 6)}\) which we explored in derivatives.
Could you please recheck the options or provide adjustments if possible, as none factor to what's solvable?