Question

Factor completely.
2, x, squared, plus, 5, x, plus, 2
2x
2
+5x+2

Answer 1

To factor the expression \( 2x^2 + 5x + 2 \) completely, we can follow these steps:

1. **Identify the coefficients**: Here, \( a = 2 \), \( b = 5 \), and \( c = 2 \).

2. **Calculate the product \( ac \)**:
\[
ac = 2 \cdot 2 = 4
\]

3. **Find two numbers that multiply to \( ac \) (4) and add to \( b \) (5)**:
The two numbers that meet these criteria are 4 and 1, since \( 4 \cdot 1 = 4 \) and \( 4 + 1 = 5 \).

4. **Rewrite the middle term using these numbers**:
\[
2x^2 + 4x + 1x + 2
\]

5. **Group the terms**:
\[
(2x^2 + 4x) + (1x + 2)
\]

6. **Factor out the common factors in each group**:
\[
2x(x + 2) + 1(x + 2)
\]

7. **Factor out the common binomial factor**:
\[
(x + 2)(2x + 1)
\]

Thus, the completely factored form of the expression \( 2x^2 + 5x + 2 \) is:
\[
\boxed{(x + 2)(2x + 1)}
\]