To factor the quadratic expression \( 2x^2 + 7x + 5 \), we will use the method of finding two numbers that multiply to \( ac \) (where \( a = 2 \) and \( c = 5 \)) and sum to \( b \) (where \( b = 7 \)).
-
First, calculate \( ac \): \[ ac = 2 \cdot 5 = 10 \]
-
We need to find two numbers that multiply to \( 10 \) and add up to \( 7 \). The numbers \( 2 \) and \( 5 \) work because: \[ 2 \cdot 5 = 10 \quad \text{and} \quad 2 + 5 = 7 \]
-
Now, we can rewrite the middle term \( 7x \) using \( 2x \) and \( 5x \): \[ 2x^2 + 2x + 5x + 5 \]
-
Next, group the terms: \[ (2x^2 + 2x) + (5x + 5) \]
-
Factor out the common factors in each group: \[ 2x(x + 1) + 5(x + 1) \]
-
Now, factor out the common binomial factor \( (x + 1) \): \[ (2x + 5)(x + 1) \]
Thus, the expression \( 2x^2 + 7x + 5 \) factors completely to: \[ \boxed{(2x + 5)(x + 1)} \]