\( x \) \( (a x+b)^{2}=4 x^{2}+p x+9 \), determine the volue of \( a, b \) and \( p \) where \( a, b, p>0 \)

To determine the values of \( a \), \( b \), and \( p \) in the equation: \[ (a x + b)^2 = 4 x^2 + p x + 9 \] where \( a, b, p > 0 \), follow these steps: 1. **Expand the Left Side:** \[ (a x + b)^2 = a^2 x^2 + 2ab x + b^2 \] 2. **Equate the Coefficients:** By comparing the expanded form with the right side of the equation: \[ a^2 x^2 + 2ab x + b^2 = 4 x^2 + p x + 9 \] This gives us a system of equations by matching the coefficients of corresponding powers of \( x \): \[ \begin{cases} a^2 = 4 \\ 2ab = p \\ b^2 = 9 \end{cases} \] 3. **Solve for \( a \) and \( b \):** - From \( a^2 = 4 \): \[ a = \sqrt{4} = 2 \quad (\text{since } a > 0) \] - From \( b^2 = 9 \): \[ b = \sqrt{9} = 3 \quad (\text{since } b > 0) \] 4. **Determine \( p \):** Using \( 2ab = p \): \[ p = 2 \times 2 \times 3 = 12 \] **Final Values:** \[ a = 2, \quad b = 3, \quad p = 12 \] **Verification:** Plugging these values back into the original equation: \[ (2x + 3)^2 = 4x^2 + 12x + 9 \] Which matches the right side of the equation, confirming the solution is correct. **Answer:** After simplifying, the positive solutions are a = 2, b = 3, and p = 12.