Complete the table. Find (i) the number of crosses needed for pattern 45, (ii) the nth term for the sum of crosses and dots.

Pattern 1: 1 dot ($1^2$)
Pattern 2: 4 dots ($2^2$)
Pattern 3: 9 dots ($3^2$)
This sequence is $n^2$. For pattern 4, the number of dots is $4^2 = 16$. This is already given in the table.

Pattern 1: 5 crosses
Pattern 2: 8 crosses
Pattern 3: 11 crosses
This is an arithmetic progression with a common difference $d = 8-5 = 3$.
The formula for the $n$-th term is $a_n = a_1 + (n-1)d$.
Here $a_1 = 5$ and $d = 3$.
So, $a_n = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2$.
For pattern 4, the number of crosses is $3(4) + 2 = 12 + 2 = 14$.

Pattern 1: $1+5 = 6$
Pattern 2: $4+8 = 12$
Pattern 3: $9+11 = 20$
For pattern 4, the sum is $16+14 = 30$.

The completed table is:

The number of crosses needed for pattern 45 is $\boxed{\text{137}}$.

6, 12, 20, ...
Find the first differences: $12-6=6$, $20-12=8$.
Find the second differences: $8-6=2$.
Since the second difference is constant, the $n$-th term is a quadratic sequence of the form $An^2 + Bn + C$.

For a quadratic sequence, $2A = \text{second difference}$, so $2A = 2 \Rightarrow A = 1$.
The first term of the first differences is $3A+B$, so $3(1)+B = 6 \Rightarrow 3+B=6 \Rightarrow B=3$.
The first term of the sequence is $A+B+C$, so $1+3+C = 6 \Rightarrow 4+C=6 \Rightarrow C=2$.

The $n$-th term for the sum of crosses and dots is $n^2 + 3n + 2$.
The $n$-th term for the sum of crosses and dots is $\boxed{n^2 + 3n + 2}$.