To calculate the surface area of the box, we can use the formula for the surface area of a rectangular prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the box. Given:
- Length \( l = 17 \) cm
- Width \( w = 10 \) cm
- Height \( h = 4 \) cm
Plugging in the values:
\[ \text{Surface Area} = 2(17 \times 10 + 17 \times 4 + 10 \times 4) \]
Calculating each term inside the parentheses:
- \( 17 \times 10 = 170 \)
- \( 17 \times 4 = 68 \)
- \( 10 \times 4 = 40 \)
Now we sum these products:
\[ 170 + 68 + 40 = 278 \]
Now multiply by 2 to find the total surface area:
\[ \text{Surface Area} = 2 \times 278 = 556 , \text{cm}^2 \]
Thus, the total amount of wrapping paper Jerry will need to wrap the present is
\[ \boxed{556} , \text{cm}^2 \]