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Choice A is correct. Let $x$ represent the side length, in inches, of square P. It follows that the perimeter of square P is $4x$ inches. It's given that square Q has a perimeter that is $176$ inches greater than the perimeter of square P. Thus, the perimeter of square Q is $176$ inches greater than $4x$ inches, or $4x+176$ inches. Since the perimeter of a square is $4$ times the side length of the square, each side length of Q is $\frac{4x+176}{4}$, or $x+44$ inches. Since the area of a square is calculated by multiplying the length of two sides, the area of square Q is $\left(x+44\right)\left(x+44\right)$, or ${\left(x+44\right)}^2$ square inches. It follows that function $f$ is defined by $f\left(x\right)={\left(x+44\right)}^2$. 

Choice B is incorrect. This function represents a square with side lengths $\left(x+176\right)$ inches.

Choice C is incorrect. This function represents a square with side lengths $\left(176x+44\right)$ inches.

Choice D is incorrect. This function represents a square with side lengths $\left(176x+176\right)$ inches.