We have the expression: (x-5)^2 + 1 = 3x - 4

(x-5)^2 + 1 = 3x - 4

open the brackets

x^2 + 25 - 10x + 1 = 3x - 4

=> x^2 - 10x - 3x + 26 + 4 = 0

=> x^2 - 13x + 30 = 0

The expression is a quadratic equation.

If we have to find the solutions

x^2 - 13x + 30 = 0

=> x^2 - 10x - 3x + 30 = 0

=> x(x - 10) - 3( x - 10) = 0

=> (x - 3)(x - 10) = 0

=> x = 3 and x = 10

**The given expression is a quadratic equation.**

For the given expression to represent a quadratic equation, it has to have the following form: ax^2 + bx + c = 0.

We'll begin by expanding the square from the left side:

(x-5)^2 = x^2 - 10x + 25

We'll re-write the equation:

x^2 - 10x + 25 + 1 = 3x - 4

We'll combine like terms from the left side:

x^2 - 10x + 26 = 3x - 4

We'll subtract 3x - 4:

x^2 - 10x + 26 - 3x + 4 = 0

We'll combine like terms:

x^2 - 13x + 30 = 0

**The final result is a quadratic equation x^2 - 13x + 30 = 0.**