3(x+4)²=10x+32

To solve the equation \( 3(x+4)^2 = 10x + 32 \), let's start by expanding and rearranging the equation.

1. Expand the left side: \[ 3(x+4)^2 = 3(x^2 + 8x + 16) = 3x^2 + 24x + 48 \]

2. Write the equation: \[ 3x^2 + 24x + 48 = 10x + 32 \]

3. Rearrange to set the equation to zero: Subtract \(10x\) and \(32\) from both sides: \[ 3x^2 + 24x + 48 - 10x - 32 = 0 \] Simplifying gives: \[ 3x^2 + 14x + 16 = 0 \]

4. Now, use the quadratic formula: The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 3\), \(b = 14\), and \(c = 16\).

5. Calculate the discriminant: \[ b^2 - 4ac = 14^2 - 4 \cdot 3 \cdot 16 = 196 - 192 = 4 \]

6. Find the roots: \[ x = \frac{-14 \pm \sqrt{4}}{2 \cdot 3} = \frac{-14 \pm 2}{6} \] This gives two possible solutions: \[ x = \frac{-14 + 2}{6} = \frac{-12}{6} = -2 \] and \[ x = \frac{-14 - 2}{6} = \frac{-16}{6} = -\frac{8}{3} \]

7. Final solutions: The solutions to the equation \( 3(x+4)^2 = 10x + 32 \) are: \[ x = -2 \quad \text{and} \quad x = -\frac{8}{3} \]