Factor the trinomial:
5, x, squared, plus, 7, x, plus, 2
5x 
2
 +7x+2

To factor the trinomial \( 5x^2 + 7x + 2 \), follow these steps:

### Step 1: Multiply the coefficient of \(x^2\) and the constant term
Multiply the coefficient of \(x^2\) (which is 5) by the constant term (which is 2):

\[
5 \times 2 = 10
\]

### Step 2: Find two numbers that multiply to 10 and add up to 7
We need two numbers that multiply to 10 and add up to 7 (the coefficient of \(x\)). The numbers 5 and 2 satisfy this:

\[
5 \times 2 = 10 \quad \text{and} \quad 5 + 2 = 7
\]

### Step 3: Rewrite the middle term using these numbers
Rewrite \(7x\) as \(5x + 2x\), so the expression becomes:

\[
5x^2 + 5x + 2x + 2
\]

### Step 4: Group the terms
Group the terms into pairs:

\[
(5x^2 + 5x) + (2x + 2)
\]

### Step 5: Factor out the greatest common factor (GCF) from each group
From the first group, factor out \(5x\), and from the second group, factor out 2:

\[
5x(x + 1) + 2(x + 1)
\]

### Step 6: Factor out the common binomial factor
Now factor out the common binomial factor \((x + 1)\):

\[
(5x + 2)(x + 1)
\]

### Final Answer:
\[
5x^2 + 7x + 2 = (5x + 2)(x + 1)
\]

Would you like a detailed explanation of any specific step?

Here are some related questions to explore:
1. How do you factor other quadratic trinomials with different coefficients?
2. What happens when a trinomial cannot be factored using integers?
3. Can you use the quadratic formula to check the factorization?
4. How do you factor trinomials when the leading coefficient is 1?
5. What are the differences between factoring and completing the square?

**Tip:** When factoring, always check if the product of the factors equals the original expression.

Learn how to factor the quadratic trinomial 5x^2 + 7x + 2 with a clear, step-by-step solution. This explanation covers factoring by grouping, finding pairs that sum to the middle term, and working with coefficients. Perfect for students in Grades 8-10 studying algebra.

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