The equation in the image is:

$x^2 - 4x - 21 = 0$

We are asked to solve this quadratic equation by factorizing.

Step-by-step solution:

1. Identify the quadratic form: The equation is in the standard quadratic form $ax^2 + bx + c = 0$, where:

   • $a = 1$ (coefficient of $x^2$),
   • $b = -4$ (coefficient of $x$),
   • $c = -21$ (constant term).

2. Find factors of the constant term $-21$: We are looking for two numbers that multiply to $-21$ and add to $-4$. The factors of $-21$ are:

   • $3$ and $-7$, because: $3 \times (-7) = -21 \quad \text{and} \quad 3 + (-7) = -4$

3. Write the factored form: Using these factors, we can rewrite the quadratic equation as: $(x + 3)(x - 7) = 0$

4. Solve for $x$: To find the solutions, set each factor equal to zero: $x + 3 = 0 \quad \text{or} \quad x - 7 = 0$ Solving these gives: $x = -3 \quad \text{or} \quad x = 7$

Final solution:

The solutions to the equation $x^2 - 4x - 21 = 0$ are $x = -3$ and $x = 7$.

Would you like further clarification on this process or more details?

Here are five related questions to deepen your understanding:

1. How do you solve a quadratic equation if it cannot be factored?
2. What is the quadratic formula, and when should you use it?
3. How can you verify the solutions to a quadratic equation?
4. What are the key characteristics of a quadratic function's graph?
5. What happens when the discriminant of a quadratic equation is negative?

Tip:

When factorizing quadratic equations, always check that the product of the factors equals the constant term and that their sum equals the coefficient of the linear term.