To determine the length of one side of Sam's square garden, we start with the expression for the area of the garden, which is given as [tex]\(4x^2 + 28x + 49\)[/tex] square feet.
Because the garden is square-shaped, the area [tex]\(A\)[/tex] of the garden can be written as:
[tex]\[ A = (\text{side length})^2 \][/tex]
We need to represent the quadratic expression [tex]\(4x^2 + 28x + 49\)[/tex] as a square of a binomial form [tex]\((ax + b)^2\)[/tex].
The quadratic expression we have is:
[tex]\[ 4x^2 + 28x + 49 \][/tex]
This can be factored into:
[tex]\[ (2x + 7)^2 \][/tex]
To verify, we can expand [tex]\((2x + 7)^2\)[/tex] and check if it matches [tex]\(4x^2 + 28x + 49\)[/tex]:
[tex]\[ (2x + 7)^2 = (2x + 7)(2x + 7) = 2x \cdot 2x + 2x \cdot 7 + 7 \cdot 2x + 7 \cdot 7 \][/tex]
[tex]\[ = 4x^2 + 14x + 14x + 49 \][/tex]
[tex]\[ = 4x^2 + 28x + 49 \][/tex]
Thus, we have correctly factored the quadratic expression. The expression inside the square represents the length of one side of the garden.
So, the length of one side of Sam's garden is [tex]\(2x + 7\)[/tex] feet.
Therefore, the correct option is:
[tex]\[ \boxed{2x + 7 \text{ feet}} \][/tex]
