The image you uploaded outlines steps to solve a quadratic equation. I will guide you through solving the equation $t^2 + 10t + 9 = 0$ step by step:

1. Identify the quadratic equation: The equation is given as $t^2 + 10t + 9 = 0$.

2. Factorize the quadratic equation: We need to find two numbers that multiply to the constant term (+9) and add to the coefficient of the linear term (+10).

   These numbers are +1 and +9, since $1 \times 9 = 9$ and $1 + 9 = 10$.

   So, the equation can be factorized as: $(t + 1)(t + 9) = 0$

3. Solve for the roots: Set each factor equal to zero: $t + 1 = 0 \quad \text{or} \quad t + 9 = 0$ Solving these gives: $t = -1 \quad \text{or} \quad t = -9$

4. Verify the solutions: Substitute $t = -1$ and $t = -9$ back into the original equation to ensure they satisfy it.

   For $t = -1$: $(-1)^2 + 10(-1) + 9 = 1 - 10 + 9 = 0$

   For $t = -9$: $(-9)^2 + 10(-9) + 9 = 81 - 90 + 9 = 0$

   Both values satisfy the original equation, so the solutions are correct.

5. Conclusion: The solutions to the equation $t^2 + 10t + 9 = 0$ are $t = -1$ and $t = -9$.

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How do you factorize a quadratic equation if it cannot be easily factored?
2. What is the quadratic formula and how is it used to solve quadratic equations?
3. Can you solve a quadratic equation by completing the square? How?
4. What are the implications of the discriminant in a quadratic equation?
5. How can quadratic equations be applied to real-world problems?

Tip: Always check if the quadratic equation can be factored easily before using more complex methods like the quadratic formula or completing the square.