Question 4 (1 point) solve. \[ 3(x+7)^{2}+17=59 \] a \( 42 \pm \sqrt{14} \) b \( -7 \pm \sqrt{14} \) c \( -7 \pm \sqrt{13} \) d \( 42 \pm \sqrt{13} \)

Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(3\left(x+7\right)^{2}+17=59\) - step1: Expand the expression: \(3x^{2}+42x+164=59\) - step2: Move the expression to the left side: \(3x^{2}+42x+105=0\) - step3: Solve using the quadratic formula: \(x=\frac{-42\pm \sqrt{42^{2}-4\times 3\times 105}}{2\times 3}\) - step4: Simplify the expression: \(x=\frac{-42\pm \sqrt{42^{2}-4\times 3\times 105}}{6}\) - step5: Simplify the expression: \(x=\frac{-42\pm \sqrt{504}}{6}\) - step6: Simplify the expression: \(x=\frac{-42\pm 6\sqrt{14}}{6}\) - step7: Separate into possible cases: \(\begin{align}&x=\frac{-42+6\sqrt{14}}{6}\\&x=\frac{-42-6\sqrt{14}}{6}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=-7+\sqrt{14}\\&x=\frac{-42-6\sqrt{14}}{6}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=-7+\sqrt{14}\\&x=-7-\sqrt{14}\end{align}\) - step10: Rewrite: \(x_{1}=-7-\sqrt{14},x_{2}=-7+\sqrt{14}\) To solve the equation \[ 3(x+7)^{2}+17=59, \] we can follow these steps: 1. **Isolate the quadratic term**: Start by subtracting 17 from both sides: \[ 3(x+7)^{2} = 59 - 17 \] This simplifies to: \[ 3(x+7)^{2} = 42. \] 2. **Divide by 3**: Next, divide both sides by 3: \[ (x+7)^{2} = \frac{42}{3} = 14. \] 3. **Take the square root**: Now, take the square root of both sides: \[ x+7 = \pm \sqrt{14}. \] 4. **Solve for \(x\)**: Finally, subtract 7 from both sides: \[ x = -7 \pm \sqrt{14}. \] Thus, the solutions are \[ x_{1} = -7 - \sqrt{14}, \quad x_{2} = -7 + \sqrt{14}. \] The correct answer corresponds to option **b**: \( -7 \pm \sqrt{14} \).