You have two sides of an equation.
Each side has a base and an exponent.
The bases are equal.
Each base is raised to an exponent.
The exponents are expressions that look different, but the sides of the equation are equal.
Then the exponents must be equal.
In math notation,
if a^m = a^n,
then m = n.
The key to this problem is to write the two sides with the same base.
Notice that 27 = 3^3.
We use the rules of exponents on the right side to write an expression with base 3.
3^(2x + 2) = 27^(x - 2)
3^(2x + 2) = (3^3)^(x - 2)
When you raise a power to a power, multiply powers.
3^(2x + 2) = 3^(3(x - 2))
3^(2x + 2) = 3^(3x - 6)
The bases are both 3. The sides are equal, so the exponents must be equal.
2x + 2 = 3x - 6
-x = -8
x = 8
Now that we know x = 8, we evaluate the expression.
x^2 + 1 = 8^2 + 1 = 64 + 1 = 65
Answer: 65
