Simplify each through multiplication/division. 9 (n^2+5n+6)/(n^2)+n-2+(n-1)/(2n-2) 10) (x^2+2x-3)/(3x-3)cdot (3x+6)/(x+3) 11) (3x^3-6x^2)/(3x^2)+9xcdot (x^2+4x+3)/(3x^{3)+3 12) (2k+4)/(k+2)div (k^2+3k+2)/(k^2)+4k+3

1. Simplify Problem 9<br /> Factor all numerators and denominators: $n^2+5n+6=(n+2)(n+3)$, $n^2+n-2=(n+2)(n-1)$, $2n-2=2(n-1)$. Rewrite: $\frac{(n+2)(n+3)}{(n+2)(n-1)}+\frac{n-1}{2(n-1)}$.<br />2. Reduce Fractions (Problem 9)<br /> Cancel $(n+2)$: $\frac{n+3}{n-1}+\frac{n-1}{2(n-1)}=\frac{n+3}{n-1}+\frac{1}{2}$.<br />3. Common Denominator (Problem 9)<br /> LCD is $2(n-1)$. Combine: $\frac{2(n+3)+1}{2(n-1)}=\frac{2n+7}{2(n-1)}$.<br /><br />4. Simplify Problem 10<br /> Factor: $x^2+2x-3=(x+3)(x-1)$, $3x-3=3(x-1)$, $3x+6=3(x+2)$. Rewrite: $\frac{(x+3)(x-1)}{3(x-1)}\cdot\frac{3(x+2)}{x+3}$.<br />5. Reduce and Multiply (Problem 10)<br /> Cancel $(x+3)$, $(x-1)$, $3$: Result is $x+2$.<br /><br />6. Simplify Problem 11<br /> Factor: $3x^3-6x^2=3x^2(x-2)$, $3x^2+9x=3x(x+3)$, $x^2+4x+3=(x+1)(x+3)$, $3x^3+3=3(x^3+1)=3(x+1)(x^2-x+1)$. Write as $\frac{3x^2(x-2)}{3x(x+3)}\cdot\frac{(x+1)(x+3)}{3(x+1)(x^2-x+1)}$.<br />7. Reduce and Multiply (Problem 11)<br /> Cancel $3$, $x+3$, $x+1$: $\frac{x(x-2)}{3(x^2-x+1)}$.<br /><br />8. Simplify Problem 12<br /> Factor: $2k+4=2(k+2)$, $k^2+3k+2=(k+1)(k+2)$, $k^2+4k+3=(k+1)(k+3)$. Division becomes multiplication by reciprocal: $\frac{2(k+2)}{k+2}\cdot\frac{(k+1)(k+3)}{(k+1)(k+2)}$.<br />9. Reduce and Multiply (Problem 12)<br /> Cancel $k+2$, $k+1$: $\frac{2(k+3)}{k+2}$.