Question: User: $(14c^2+91c+44)\div(c+6)$
1. **State the problem:** Simplify the expression $$\frac{14c^2+91c+44}{c+6}$$.
2. **Formula and rules:** To simplify a rational expression, factor the numerator and denominator if possible, then cancel common factors.
3. **Factor the numerator:**
$$14c^2+91c+44$$
Look for two numbers that multiply to $$14 \times 44 = 616$$ and add to $$91$$.
These numbers are $$77$$ and $$8$$.
Rewrite the middle term:
$$14c^2 + 77c + 8c + 44$$
Group terms:
$$ (14c^2 + 77c) + (8c + 44) $$
Factor each group:
$$7c(2c + 11) + 4(2c + 11)$$
Factor out the common binomial:
$$ (7c + 4)(2c + 11) $$
4. **Factor the denominator:**
$$c + 6$$ (cannot be factored further).
5. **Check for common factors:**
No common factors between numerator and denominator.
6. **Final simplified expression:**
$$\frac{(7c + 4)(2c + 11)}{c + 6}$$
Since no factors cancel, this is the simplified form.