Let's transform each quadratic equation into the standard form $y = ax^2 + bx + c$.

1) $y - x + 5 = (x - 2)^2$

Expand $(x - 2)^2$: $(x - 2)^2 = x^2 - 4x + 4$ So the equation becomes: $y - x + 5 = x^2 - 4x + 4$ Solve for $y$: $y = x^2 - 4x + 4 + x - 5$ Simplify: $y = x^2 - 3x - 1$ Thus, in the form $y = ax^2 + bx + c$, we have: $y = x^2 - 3x - 1$

2) $x + 5 = (x + 1)^2 - y$

Expand $(x + 1)^2$: $(x + 1)^2 = x^2 + 2x + 1$ So the equation becomes: $x + 5 = x^2 + 2x + 1 - y$ Solve for $y$: $y = x^2 + 2x + 1 - x - 5$ Simplify: $y = x^2 + x - 4$ Thus, in the form $y = ax^2 + bx + c$, we have: $y = x^2 + x - 4$

3) $4x = x^2 + 4 - y$

Solve for $y$: $y = x^2 + 4 - 4x$ Rearrange: $y = x^2 - 4x + 4$ Thus, in the form $y = ax^2 + bx + c$, we have: $y = x^2 - 4x + 4$

4) $y - x = x^2 - x + 1$

Solve for $y$: $y = x^2 - x + 1 + x$ Simplify: $y = x^2 + 1$ Thus, in the form $y = ax^2 + bx + c$, we have: $y = x^2 + 1$

5) $4x - y = x + 5$

Solve for $y$: $y = 4x - x - 5$ Simplify: $y = 3x - 5$ Thus, in the form $y = ax^2 + bx + c$, we have: $y = 0x^2 + 3x - 5$

Final Answers:

1. $y = x^2 - 3x - 1$
2. $y = x^2 + x - 4$
3. $y = x^2 - 4x + 4$
4. $y = x^2 + 1$
5. $y = 3x - 5$

---

Would you like further details or explanations on any of the steps?

Here are 5 related questions to consider:

1. How do you complete the square for any quadratic equation?
2. What are the general forms of a quadratic equation?
3. How can you determine the vertex from a quadratic equation?
4. How does factoring help in solving quadratic equations?
5. What are the properties of a parabola described by $y = ax^2 + bx + c$?

Tip: Always simplify and rearrange terms carefully when transforming equations to avoid mistakes.