To find the value of \( x \), we start with the formula for the perimeter of a rectangle, which is given by:
\[ P = 2(\text{length} + \text{width}) \]
In this case, the length is \( 5x + 17 \) and the width is \( 3x + 21 \). Therefore, we can express the perimeter as:
\[ 196 = 2((5x + 17) + (3x + 21)) \]
First, simplify the expression inside the parentheses:
\[ (5x + 17) + (3x + 21) = 5x + 3x + 17 + 21 = 8x + 38 \]
Then, insert this back into the perimeter formula:
\[ 196 = 2(8x + 38) \]
Now, divide both sides by 2 to simplify:
\[ 98 = 8x + 38 \]
Next, subtract 38 from both sides:
\[ 98 - 38 = 8x \] \[ 60 = 8x \]
Now, divide both sides by 8 to solve for \( x \):
\[ x = \frac{60}{8} = 7.5 \]
Thus, the value of \( x \) is:
\[ \boxed{7.5} \]