Express the product of $$\frac{x+2}{2}$$ and $$\frac{4x+20}{x^2+6x+8}$$ in simplest form

$$\frac{x+2}{2}\cdot\frac{4x+20}{x^2+6x+8}=\\ \frac{x+2}{2}\cdot\frac{2(2x+10)}{x^2+2x+4x+8}=\\ (x+2)\cdot\frac{2x+10}{x(x+2)+4(x+2)}=\\ (x+2)\cdot\frac{2x+10}{(x+4)(x+2)}=\\ \frac{2x+10}{x+4}$$

$$\frac{ x+2 }{2} \cdot \frac{ 4x+20 }{x^2+6x+8 } \\\\x^2+6x+8\neq 0\\ \\x^2+4x +2x+8 \neq 0 \\ \\ x(x+4)+2(x+4)\neq 0\\ \\(x+4)(x+2)\neq 0$$

$$x+4 \neq 0 \ \ \wedge \ \ x+2\neq 0\\ \\ x\neq -4 \ \ \wedge \ \ x\neq -2 \\ \\ D=R\setminus \left \{ -4,2 \right \}$$

$$\frac{ x+2 }{2} \cdot \frac{ 4x+20 }{x^2+6x+8 } =\frac{\not( x+2 )^1}{\not2^1} \cdot \frac{ \not4^2(x+5) }{\not(x+2)^1(x+4) } =\frac{2(x+5)}{x+4}$$

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