Question

Simplify completely quantity x squared plus 12 x plus 35 all over 3 x plus 15.

quantity of x squared plus 21 over 3
quantity of x squared plus 11 over 3
quantity x plus 7 over 3
quantity of x plus 5 over 3

Answer 1

To simplify the expression \(\frac{x^2 + 12x + 35}{3x + 15}\), we will first factor both the numerator and the denominator.

1. **Factor the numerator \(x^2 + 12x + 35\)**:
We look for two numbers that multiply to \(35\) (the constant term) and add to \(12\) (the coefficient of \(x\)). The numbers \(5\) and \(7\) work because:
\[
5 \times 7 = 35 \quad \text{and} \quad 5 + 7 = 12.
\]
Thus, we can factor the numerator as:
\[
x^2 + 12x + 35 = (x + 5)(x + 7).
\]

2. **Factor the denominator \(3x + 15\)**:
We can factor out a \(3\):
\[
3x + 15 = 3(x + 5).
\]

3. **Rewrite the original expression using the factored forms**:
\[
\frac{x^2 + 12x + 35}{3x + 15} = \frac{(x + 5)(x + 7)}{3(x + 5)}.
\]

4. **Cancel the common factor \((x + 5)\)**, assuming \(x \neq -5\):
\[
= \frac{x + 7}{3}.
\]

Thus, the simplified form of your original expression is:
\[
\frac{x + 7}{3}.
\]

**Final Answer:** \(\frac{x + 7}{3}\)