The area of a square is \( \left(9 x^{2}+30 x+\mathrm{m}\right) \mathrm{cm}^{2} \). Find the value of m (given that the side length of the square is a rational number), then find its perimeter when \( x=2 \)

To find the value of \( m \) such that the area of the square, given by \( 9x^2 + 30x + m \), is a perfect square (since the side length of the square must be a rational number), we start by expressing the area as a perfect square trinomial. A perfect square trinomial can be expressed in the form \( (ax + b)^2 \), which expands to: \[ a^2x^2 + 2abx + b^2 \] Comparing this with the given expression \( 9x^2 + 30x + m \), we can identify: - \( a^2 = 9 \) implies \( a = 3 \) - \( 2ab = 30 \) From \( a = 3 \), we can substitute into the second equation: \[ 2(3)b = 30 \implies 6b = 30 \implies b = 5 \] Now, we can find \( b^2 \): \[ b^2 = 5^2 = 25 \] Thus, we have \( m = b^2 = 25 \). Now, substituting \( m \) back into the area expression, we have: \[ 9x^2 + 30x + 25 = (3x + 5)^2 \] This confirms that the area is indeed a perfect square, and the side length of the square is \( 3x + 5 \). Next, we need to find the perimeter of the square when \( x = 2 \): First, calculate the side length: \[ 3(2) + 5 = 6 + 5 = 11 \text{ cm} \] The perimeter \( P \) of a square is given by: \[ P = 4 \times \text{side length} = 4 \times 11 = 44 \text{ cm} \] Thus, the value of \( m \) is \( 25 \) and the perimeter when \( x = 2 \) is \( \boxed{44} \) cm.