Factor completely.

Question

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: \[ \boxed{(x + 2)(2x + 1)} \]