Let's solve the equation step by step.

The given equation is:

$x^2 - 10x + 21 = x - 3$

Step 1: Move all terms to one side

We can subtract $x$ and add 3 from both sides to set the equation to 0:

$x^2 - 10x + 21 - x + 3 = 0$

This simplifies to:

$x^2 - 11x + 24 = 0$

Step 2: Factor the quadratic equation

Now, we need to factor $x^2 - 11x + 24$. We are looking for two numbers that multiply to 24 and add up to -11. These numbers are -3 and -8. So we can factor the quadratic as:

$(x - 3)(x - 8) = 0$

Step 3: Solve for $x$

Now, set each factor equal to zero and solve for $x$:

1. $x - 3 = 0 \Rightarrow x = 3$
2. $x - 8 = 0 \Rightarrow x = 8$

Final Answer:

The solutions are:

$x = 3 \quad \text{or} \quad x = 8$

Do you want more details or have any questions?

Here are some related questions for further practice:

1. Solve: $x^2 - 7x + 12 = x$
2. Solve: $x^2 - 6x + 9 = 0$
3. Solve: $x^2 - 5x - 14 = 0$
4. Solve: $2x^2 - 9x + 10 = x + 5$
5. Solve: $x^2 - 8x + 15 = 0$

Tip: When factoring quadratic equations, always look for pairs of numbers whose product equals the constant term and whose sum equals the coefficient of the middle term.