Certainly! Let's factor the given trinomial step-by-step.
We are given the trinomial:
[tex]\[ 2x^2 + 5x + 2 \][/tex]
### Step 1: Find two numbers that multiply to [tex]\((a \cdot c)\)[/tex] and add to [tex]\(b\)[/tex]
First, identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic expression [tex]\(ax^2 + bx + c\)[/tex].
In this case:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 5\)[/tex]
- [tex]\(c = 2\)[/tex]
We need to find two numbers that multiply to [tex]\(a \cdot c = 2 \cdot 2 = 4\)[/tex] and add up to [tex]\(b = 5\)[/tex].
Those two numbers are [tex]\(4\)[/tex] and [tex]\(1\)[/tex], because:
[tex]\[ 4 \cdot 1 = 4 \][/tex]
[tex]\[ 4 + 1 = 5 \][/tex]
### Step 2: Split the middle term using the two numbers found
We can rewrite the middle term [tex]\(5x\)[/tex] as [tex]\(4x + 1x\)[/tex] to split the expression:
[tex]\[ 2x^2 + 5x + 2 = 2x^2 + 4x + x + 2 \][/tex]
### Step 3: Factor by grouping
Now, group the terms in pairs and factor out the common factor from each pair:
[tex]\[ 2x^2 + 4x + x + 2 \][/tex]
Group the first two terms and the last two terms:
[tex]\[ (2x^2 + 4x) + (x + 2) \][/tex]
Factor out the greatest common factor from each group:
[tex]\[ 2x(x + 2) + 1(x + 2) \][/tex]
### Step 4: Factor out the common binomial factor
Now, notice that [tex]\((x + 2)\)[/tex] is a common factor:
[tex]\[ (2x + 1)(x + 2) \][/tex]
So, the factored form of the trinomial [tex]\(2x^2 + 5x + 2\)[/tex] is:
[tex]\[ (x + 2)(2x + 1) \][/tex]
This is the correct factored form. Therefore, the final answer is:
[tex]\[ (x + 2)(2x + 1) \][/tex]
