1. **State the problem:** Given two functions $f(x) = x^2 + 5x + 6$ and $g(x) = x + 3$, find the following combined functions: - $(f + g)(x)$ - $(f - g)(x)$ - $(f \cdot g)(x)$ - $\left(\frac{f}{g}\right)(x)$ 2. **Recall the formulas:** - Sum: $(f + g)(x) = f(x) + g(x)$ - Difference: $(f - g)(x) = f(x) - g(x)$ - Product: $(f \cdot g)(x) = f(x) \times g(x)$ - Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$ 3. **Calculate each:** - $(f + g)(x) = (x^2 + 5x + 6) + (x + 3) = x^2 + 5x + 6 + x + 3 = x^2 + 6x + 9$ - $(f - g)(x) = (x^2 + 5x + 6) - (x + 3) = x^2 + 5x + 6 - x - 3 = x^2 + 4x + 3$ - $(f \cdot g)(x) = (x^2 + 5x + 6)(x + 3)$ Multiply each term: $$x^2 \times x = x^3$$ $$x^2 \times 3 = 3x^2$$ $$5x \times x = 5x^2$$ $$5x \times 3 = 15x$$ $$6 \times x = 6x$$ $$6 \times 3 = 18$$ Sum all: $$x^3 + 3x^2 + 5x^2 + 15x + 6x + 18 = x^3 + 8x^2 + 21x + 18$$ - $\left(\frac{f}{g}\right)(x) = \frac{x^2 + 5x + 6}{x + 3}$ Factor numerator: $$x^2 + 5x + 6 = (x + 2)(x + 3)$$ So: $$\frac{(x + 2)(x + 3)}{x + 3}$$ Cancel common factor $x + 3$ (where $x \neq -3$): $$\frac{\cancel{(x + 3)}(x + 2)}{\cancel{(x + 3)}} = x + 2$$ 4. **Final answers:** - $(f + g)(x) = x^2 + 6x + 9$ - $(f - g)(x) = x^2 + 4x + 3$ - $(f \cdot g)(x) = x^3 + 8x^2 + 21x + 18$ - $\left(\frac{f}{g}\right)(x) = x + 2$, with $x \neq -3$ These results show how to combine functions by addition, subtraction, multiplication, and division, applying algebraic operations and factoring to simplify expressions.