1. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} g(3)&=&(3)^3+5(3)^2 \\ &=&27+45 \\ &=&72 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} f(3)&=&2(3)+4 \\ &=&6+4 \\ &=&10 \\ \end{array} \end{array}$
$(3)+f(3)=72+10=82$
2. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} f(-4)&=&-3(-4)^2+3(-4) \\ &=&-3(16)-12 \\ &=&-48-12 \\ &=&-60 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} g(-4)&=&2(-4)+5 \\ &=&-8+5 \\ &=&-3\\ \end{array} \end{array}$
$\dfrac{f(-4)}{g(-4)}=\dfrac{-60}{-3}=20$
3. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} g(5)&=&-4(5)+1 \\ &=&-20+1 \\ &=&-19 \end{array} & \hspace{0.25in} \begin{array}[t]{rrl} h(5)&=&-2(5)-1 \\ &=&-10-1 \\ &=& -11\\ \end{array} \end{array}$
$g(5)+h(5)=-19-11=-30$
4. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} g(2)&=&3(2)+1 \\ &=&6+1 \\ &=&7 \end{array} & \hspace{0.25in} \begin{array}[t]{rrl} f(2)&=&(2)^3+3(2)^2 \\ &=&8+3\cdot 4 \\ &=&8+12 \\ &=&20\\ \end{array} \end{array}$
$g(2)\cdot f(2)=7\cdot 20=140$
5. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} g(1)&=&1-3 \\ &=&-2 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} h(1)&=&-3(1)^3+6(1) \\ &=&-3+6 \\ &=&3\\ \end{array} \end{array}$
$g(1)+h(1)=-2+3=1$
6. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} g(-6)&=&(-6)^2-2 \\ &=&36-2 \\ &=&34 \end{array} & \hspace{0.25in} \begin{array}[t]{rrl} h(-6)&=&2(-6)+5 \\ &=&-12+5 \\ &=&-7\\ \end{array} \end{array}$
$g(-6)+h(-6)=34-7=27$
7. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} h(0)&=&\cancel{2(0)}-1 \\ &=&-1 \end{array} & \hspace{0.25in} \begin{array}[t]{rrl} g(0)&=&\cancel{3(0)}-5 \\ &=&-5 \end{array} \end{array}$
$\dfrac{h(0)}{g(0)}=\dfrac{-1}{-5}=\dfrac{1}{5}$
8. $\phantom{a}$
$(g+h)= \begin{array}[t]{rrrr} &3a&-&2 \\ +&4a&-&2 \\ \hline &7a&-&4 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g+h)(10)&=&7(10)-4 \\ &=&70-4 \\ &=&66 \end{array}$
9. $\phantom{a}$
$(g+f)= \begin{array}[t]{rrrr} &3a&+&3 \\ +&2a&-&2 \\ \hline &5a&+&1 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g+f)(9)&=&5(9)+1 \\ &=&45+1 \\ &=&46 \end{array}$
10. $\phantom{a}$
$(g-h)= \begin{array}[t]{r} 4x+3 \\ - \hspace{0.42in} (x^3-2x^2) \\ \hline -x^3+2x^2+4x+3 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g-h)(-1)&=&-(-1)^3+2(-1)^2+4(-1)+3 \\ &=&1+2-4+3 \\ &=&2 \end{array}$
11. $\phantom{a}$
$(g-f)= \begin{array}[t]{rrrr} &x&+&3 \\ -&(-x&+&4) \\ \hline &2x&-&1 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g-f)(3)&=&2(3)-1 \\ &=&6-1 \\ &=&5 \end{array}$
12. $\phantom{a}$
$(g-f)= \begin{array}[t]{rrrrrr} &x^2&&&+&2 \\ -&&&(2x&+&5) \\ \hline &x^2&-&2x&-&3 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g-f)(0)&=&\cancel{(0)^2}-\cancel{2(0)}-3 \\ &=&-3 \end{array}$
13. $\phantom{a}$
$(f+g)= \begin{array}[t]{rrrr} &n&-&5 \\ +&4n&+&2 \\ \hline &5n&-&3 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (f+g)(-8)&=&5(-8)-3 \\ &=&-40-3 \\ &=&-43 \end{array}$
14. $\phantom{a}$
$(h\cdot g)= \begin{array}[t]{rrrrrr} &&&t&+&5 \\ \times &&&3t&-&5 \\ \hline &3t^2&+&15t&& \\ &&-&5t&-&25 \\ \hline &3t^2&+&10t&-&25 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (h\cdot g)(5)&=&3(5)^2+10(5)-25 \\ &=&75+50-25 \\ &=&100 \end{array}$
15. $\phantom{a}$
$(g\cdot h)= \begin{array}[t]{rrrr} &t&-&4 \\ \times &&&2t \\ \hline &2t^2&-&8t \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g\cdot h)(3t)&=&2(3t)^2-8(3t) \\ &=&2(9t^2)-24t \\ &=&18t^2-24t \end{array}$
16. $\dfrac{g(n)}{f(n)}=\dfrac{n^2+5}{2n+5}\hspace{0.25in} \text{Does not reduce}$
17. $\dfrac{g}{f}=\dfrac{-2a+5}{3a+5}\hspace{0.3in}\left(\dfrac{g}{f}\right)(a^2)=\dfrac{-2a^2+5}{3a^2+5}\hspace{0.25in} \text{Does not reduce}$
18. $\phantom{a}$
$h(n)+g(n)= \begin{array}[t]{rrrrrr} &n^3&+&4n&& \\ +&&&4n&+&5 \\ \hline &n^3&+&8n&+&5 \end{array}$
19. $g(n^2)=(n^2)^2-4(n^2)$
$\phantom{g(n^2)}=n^4-4n^2 \hspace{1in} h(n^2)=n^2-5$
$g(n^2)\cdot h(n^2)= \begin{array}[t]{rrrrrr} &&&n^4&-&4n^2 \\ \times&&&n^2&-&5 \\ \hline &n^6&-&4n^4&& \\ &&-&5n^4&+&20n^2 \\ \hline &n^6&-&9n^4&+&20n^2 \\ \end{array}$
20. $\phantom{a}$
$(g\cdot h)= \begin{array}[t]{rrrrrr} &&&n&+&5 \\ \times &&&2n&-&5 \\ \hline &2n^2&+&10n&& \\ &&-&5n&-&25 \\ \hline &2n^2&+&5n&-&25 \end{array}\hspace{0.25in} \begin{array}[t]{rrl} (g\cdot h)(-3n)&=&2(-3n)^2+5(-3n)-25 \\ &=&2(9n^2)-15n-25 \\ &=&18n^2-15n-25 \end{array}$
21. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} (f\circ g)&=&-4(4x+3)+1 \\ &=&-16x-12+1 \\ &=&-16x-11 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} (f\circ g)(9)&=&-16(9)-11 \\ &=&-144-11 \\ &=&-155 \end{array} \end{array}$
22. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} (h\circ g)&=&3(a+1)+3 \\ &=&3a+3+3 \\ &=&3a+6 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} (h\circ g)(5)&=&3(5)+6 \\ &=&15+6 \\ &=&21 \end{array} \end{array}$
23. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} (g\circ h)&=&(x^2-1)+4 \\ &=&x^2-1+4 \\ &=&x^2+3 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} (g\circ h)(10)&=&(10)^2+3 \\ &=&100+3 \\ &=&103 \end{array} \end{array}$
24. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} (f\circ g)&=&-4(n+4)+2 \\ &=&-4n-16+2 \\ &=&-4n-14 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} (f\circ g)(9)&=&-4(9)-14 \\ &=&-36-14 \\ &=&-50 \end{array} \end{array}$
25. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrl} (g\circ h)&=&2(2x^3+4x^2)-4 \\ &=&4x^3+8x^2-4 \end{array} &\hspace{0.25in} \begin{array}[t]{rrl} (g\circ h)(3)&=&4(3)^3+8(3)^2-4 \\ &=&108+72-4 \\ &=&176 \end{array} \end{array}$
26. $(g\circ h)=(4x+4)^2-5(4x+4)$
$\phantom{g\circ h)}=16x^2+32x+16-20x-20$
$\phantom{g\circ h)}=16x^2+12x-4$
27. $(f\circ g)=-2(4a)+2$
$\phantom{f\circ g)}=-8a+2$
28. $(g\circ f)=4(x^3-1)+4$
$\phantom{g\circ f)}=4x^3-4+4$
$\phantom{g\circ f)}=4x^3$
29. $(g\circ f)=-(2x-3)+5$
$\phantom{g\circ f)}=-2x+6+5$
$\phantom{g\circ f)}=-2x+11$
30. $(f\circ g)=4(-4t-2)+3$
$\phantom{f\circ g)}=-16t-8+3$
$\phantom{f\circ g)}=-16t-5$