What are the solutions to `\left(x+7\right)^{2}=81?`

To solve the equation \((x + 7)^{2} = 81\), we will start by taking the square root of both sides.

\[ x + 7 = \pm 9 \]

This gives us two separate equations to solve:

1. \(x + 7 = 9\)
2. \(x + 7 = -9\)

Solving the first equation:

\[ x + 7 = 9 \] Subtracting 7 from both sides: \[ x = 9 - 7 \] \[ x = 2 \]

Solving the second equation:

\[ x + 7 = -9 \] Subtracting 7 from both sides: \[ x = -9 - 7 \] \[ x = -16 \]

Thus, the solutions to the equation \((x + 7)^{2} = 81\) are

\[ \boxed{2} \quad \text{and} \quad \boxed{-16}. \]

To solve the equation \(2(x - 7)^{2} = 32\), we first divide both sides by 2:

\[ (x - 7)^{2} = \frac{32}{2} \]

\[ (x - 7)^{2} = 16 \]

Next, we take the square root of both sides:

\[ x - 7 = \pm 4 \]

This gives us two separate equations to solve:

1. \(x - 7 = 4\)
2. \(x - 7 = -4\)

Solving the first equation:

\[ x - 7 = 4 \] Adding 7 to both sides: \[ x = 4 + 7 \] \[ x = 11 \]

Solving the second equation:

\[ x - 7 = -4 \] Adding 7 to both sides: \[ x = -4 + 7 \] \[ x = 3 \]

Thus, the solutions to the equation \(2(x - 7)^{2} = 32\) are

\[ \boxed{11} \quad \text{and} \quad \boxed{3}. \]