Show me the steps for factoring the equation 7x^2+28x+21

To factor the quadratic equation \( 7x^2 + 28x + 21 \), follow these steps:

### Step 1: Identify the greatest common factor (GCF)
Look for the greatest common factor (GCF) of all the terms. In this case, the GCF of \( 7x^2 \), \( 28x \), and \( 21 \) is \( 7 \).

\[
7x^2 + 28x + 21 = 7(x^2 + 4x + 3)
\]

### Step 2: Factor the quadratic inside the parentheses
Now, we focus on factoring the quadratic expression \( x^2 + 4x + 3 \).

We need two numbers that multiply to \( 3 \) (the constant term) and add up to \( 4 \) (the coefficient of \( x \)). These numbers are \( 3 \) and \( 1 \), since:

\[
3 \times 1 = 3 \quad \text{and} \quad 3 + 1 = 4
\]

Thus, we can factor the quadratic as:

\[
x^2 + 4x + 3 = (x + 3)(x + 1)
\]

### Step 3: Rewrite the expression with the factors
Now, substitute the factored form of the quadratic back into the original expression:

\[
7(x^2 + 4x + 3) = 7(x + 3)(x + 1)
\]

### Final factored form
The fully factored form of the given quadratic equation is:

\[
7x^2 + 28x + 21 = 7(x + 3)(x + 1)
\]

Would you like more details on any step or have any questions?

Here are 5 related questions to further explore this concept:
1. How would you factor \( 6x^2 + 18x + 12 \)?
2. Can you factor the equation \( 8x^2 + 16x - 20 \)?
3. What are the roots of the equation \( 7(x + 3)(x + 1) = 0 \)?
4. How can you factor a trinomial when the leading coefficient is not 1?
5. How can you use the quadratic formula to solve \( 7x^2 + 28x + 21 = 0 \)?

**Tip:** Always look for the greatest common factor (GCF) before attempting to factor trinomials, as it simplifies the process.

Learn how to factor the quadratic equation 7x^2 + 28x + 21 with step-by-step guidance. This problem involves finding the GCF and factoring a trinomial, suitable for students in Grades 8-10.

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