Evaluate 3x3 + 4x - 5 given that x = 1.

$$
3(1)^2 + 4(1) - 5
$$

$$
3(1) + 4 - 5
$$
$$
3 + 4 - 5
$$
$$
7 - 5 = 2
$$
The value of the expression is $\boxed{2}$.

$$
\frac{1}{2} + \frac{2}{2(2)+1}
$$

$$
\frac{1}{2} + \frac{2}{4+1}
$$
$$
\frac{1}{2} + \frac{2}{5}
$$

The common denominator for 2 and 5 is 10.
$$
\frac{1 \times 5}{2 \times 5} + \frac{2 \times 2}{5 \times 2}
$$
$$
\frac{5}{10} + \frac{4}{10}
$$
$$
\frac{5+4}{10} = \frac{9}{10}
$$
The value of the expression is $\boxed{\frac{9}{10}}$.

$$
\frac{1+2}{2(1)+2} + \frac{1}{2}
$$

$$
\frac{3}{2+2} + \frac{1}{2}
$$
$$
\frac{3}{4} + \frac{1}{2}
$$

The common denominator for 4 and 2 is 4.
$$
\frac{3}{4} + \frac{1 \times 2}{2 \times 2}
$$
$$
\frac{3}{4} + \frac{2}{4}
$$
$$
\frac{3+2}{4} = \frac{5}{4}
$$
The value of the expression is $\boxed{\frac{5}{4}}$.

$$
\frac{10ab}{9} \times \frac{27}{5abc} = \frac{10 \times 27 \times ab}{9 \times 5 \times abc}
$$
Cancel numerical factors: $10/5 = 2$, $27/9 = 3$.  
Cancel variable factors: $ab/ab = 1$.
$$
\frac{2 \times 3}{c} = \frac{6}{c}
$$

$$
\frac{6}{2} = 3
$$
The simplified value is $\boxed{3}$.

$$
\frac{3ab^2}{8xy} \times \frac{16xy^3}{21ab^2} = \frac{3 \times 16 \times ab^2 \times xy^3}{8 \times 21 \times xy \times ab^2}
$$
Cancel numerical factors: $3/21 = 1/7$, $16/8 = 2$.  
Cancel variable factors: $ab^2/ab^2 = 1$, $xy/xy = 1$, $y^3/y = y^2$.
$$
\frac{1 \times 2 \times y^2}{7 \times 1} = \frac{2y^2}{7}
$$

$$
\frac{4x(a+2)}{3x-1} + \frac{a+2}{2(3x-1)}
$$

$$
\frac{4x(a+2) \times 2}{2(3x-1)} + \frac{a+2}{2(3x-1)}
$$
$$
\frac{8x(a+2) + (a+2)}{2(3x-1)}
$$

$$
\frac{(a+2)(8x+1)}{2(3x-1)}
$$

$$
\frac{(2+2)(8(-1)+1)}{2(3(-1)-1)}
$$
$$
\frac{(4)(-8+1)}{2(-3-1)}
$$
$$
\frac{(4)(-7)}{2(-4)}
$$
$$
\frac{-28}{-8}
$$

$$
\frac{28}{8} = \frac{7}{2}
$$
The simplified value is $\boxed{\frac{7}{2}}$.

$$
\frac{36m^2n^2}{8m} = \frac{36}{8} \times \frac{m^2}{m} \times n^2 = \frac{9}{2}mn^2
$$

$$
\frac{n^3m^2}{4mn^2} = \frac{1}{4} \times \frac{m^2}{m} \times \frac{n^3}{n^2} = \frac{1}{4}mn
$$

$$
\frac{9}{2}mn^2 + \frac{1}{4}mn
$$

$$
\frac{9}{2}(3)\left(\frac{1}{3}\right)^2 + \frac{1}{4}(3)\left(\frac{1}{3}\right)
$$
$$
\frac{9}{2}(3)\left(\frac{1}{9}\right) + \frac{1}{4}(3)\left(\frac{1}{3}\right)
$$
$$
\frac{27}{18} + \frac{3}{12}
$$

$$
\frac{3}{2} + \frac{1}{4}
$$

$$
\frac{3 \times 2}{2 \times 2} + \frac{1}{4} = \frac{6}{4} + \frac{1}{4} = \frac{7}{4}
$$
The simplified value is $\boxed{\frac{7}{4}}$.

$$
\frac{3x(x+2)}{2y(x+2)} \times \frac{2y}{9x^2}
$$

Cancel $(x+2)$ from numerator and denominator.  
Cancel $2y$ from numerator and denominator.  
Cancel $3x$ from $3x$ in the numerator and $9x^2$ in the denominator ($3x/9x^2 = 1/3x$).
$$
\frac{1}{1} \times \frac{1}{3x} = \frac{1}{3x}
$$