To solve the quadratic equation $x^2 + 4x + 3 = 0$, we use the factorization method or the quadratic formula. Let's solve it step by step using factorization:

Step 1: Identify coefficients

The equation is in the form $ax^2 + bx + c = 0$, where:

• $a = 1$
• $b = 4$
• $c = 3$

Step 2: Find two numbers that multiply to $a \cdot c = 1 \cdot 3 = 3$ and add to $b = 4$.

These numbers are $3$ and $1$, because: $3 \cdot 1 = 3 \quad \text{and} \quad 3 + 1 = 4.$

Step 3: Rewrite the middle term

We rewrite $4x$ as $3x + 1x$: $x^2 + 3x + 1x + 3 = 0$

Step 4: Factorize by grouping

Group the terms into two pairs: $(x^2 + 3x) + (1x + 3) = 0$

Factor out the common terms from each group: $x(x + 3) + 1(x + 3) = 0$

Factor out the common binomial factor: $(x + 3)(x + 1) = 0$

Step 5: Solve for $x$

Set each factor to 0:

1. $x + 3 = 0 \quad \Rightarrow \quad x = -3$
2. $x + 1 = 0 \quad \Rightarrow \quad x = -1$

Final Answer:

The solutions are: $x = -3 \quad \text{and} \quad x = -1$

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Would you like further details, or have any questions? Here are 5 related questions to explore:

1. How can we solve this equation using the quadratic formula?
2. What are the properties of quadratic equations?
3. How do you determine if a quadratic equation has real or complex roots?
4. Can you explain the graph of this quadratic equation and its vertex?
5. What are other methods to factorize quadratic equations?

Tip: Always verify your solutions by substituting them back into the original equation!