O/N 17/P22/Q5/b,c (a) Solve \( 2 x(x+1)=3(4-x) \).

To solve the equation \( 2x(x+1) = 3(4-x) \), first expand both sides: Left side: \( 2x^2 + 2x \) Right side: \( 12 - 3x \) Now, set the equation to zero: \[ 2x^2 + 2x + 3x - 12 = 0 \] Combine like terms: \[ 2x^2 + 5x - 12 = 0 \] Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 5 \), and \( c = -12 \). Calculate the discriminant: \[ b^2 - 4ac = 5^2 - 4(2)(-12) = 25 + 96 = 121 \] Now apply the quadratic formula: \[ x = \frac{-5 \pm \sqrt{121}}{2(2)} = \frac{-5 \pm 11}{4} \] This yields two solutions: \[ x = \frac{6}{4} = \frac{3}{2} \quad \text{and} \quad x = \frac{-16}{4} = -4 \] Thus, the solutions to the equation are \( x = \frac{3}{2} \) and \( x = -4 \).