find and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$, $h\neq0$ for the given function.\n2)…

Explanation:

Step1: Find f(x + h)

Substitute (x+h) into (f(x)): [f(x + h)=(x + h)^2+6(x + h)+4=x^{2}+2xh+h^{2}+6x + 6h+4]

Step2: Substitute f(x + h) and f(x) into the difference - quotient formula

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(x^{2}+2xh+h^{2}+6x + 6h + 4)-(x^{2}+6x + 4)}{h}\ &=\frac{x^{2}+2xh+h^{2}+6x + 6h + 4 - x^{2}-6x - 4}{h} \end{align*} ]

Step3: Simplify the numerator

Combine like - terms in the numerator: [ \begin{align*} \frac{x^{2}-x^{2}+2xh+h^{2}+6x-6x + 6h+4 - 4}{h}&=\frac{2xh+h^{2}+6h}{h} \end{align*} ]

Step4: Factor out h from the numerator and simplify

Factor out (h) from the numerator: (\frac{h(2x + h+6)}{h}). Since (h\neq0), cancel out the (h) terms. [2x+h + 6]

Answer:

(2x+h + 6)