Factor the polynomial $2 x^2+7 x+6$.

Find two numbers that multiply to 2 ⋅ 6 = 12 and add up to 7 (3 and 4).
Rewrite the middle term: 2 x 2 + 3 x + 4 x + 6 .
Factor by grouping: x ( 2 x + 3 ) + 2 ( 2 x + 3 ) .
Factor out the common binomial: ( 2 x + 3 ) ( x + 2 ) . The factored form is ( 2 x + 3 ) ( x + 2 ) ​ .

Explanation

Understanding the Problem We are given the quadratic polynomial 2 x 2 + 7 x + 6 and asked to factor it.

Finding the Right Numbers We need to find two binomials that multiply to give the quadratic polynomial. We look for two numbers whose product is 2 × 6 = 12 and whose sum is 7. These numbers are 3 and 4.

Rewriting the Middle Term We rewrite the middle term using these numbers: 2 x 2 + 3 x + 4 x + 6 .

Factoring by Grouping Now, we factor by grouping: x ( 2 x + 3 ) + 2 ( 2 x + 3 ) .

Factoring out the Common Factor Finally, we factor out the common binomial factor: ( 2 x + 3 ) ( x + 2 ) . Therefore, the factored form of the polynomial is ( 2 x + 3 ) ( x + 2 ) .

Final Answer Thus, the factored form of the quadratic polynomial 2 x 2 + 7 x + 6 is ( 2 x + 3 ) ( x + 2 ) .


Examples
Factoring quadratic polynomials is a fundamental skill in algebra. It's used in many real-world applications, such as determining the dimensions of a rectangular garden given its area, or in physics to solve projectile motion problems. For example, if the area of a rectangular garden is represented by the expression 2 x 2 + 7 x + 6 , where x is a variable related to the dimensions, factoring this expression into ( 2 x + 3 ) ( x + 2 ) allows you to find possible expressions for the length and width of the garden. This skill is also crucial in more advanced mathematical concepts like calculus and differential equations.

Answered by GinnyAnswer | 2025-07-06