Factor: $x^2+4 x+3$
A. $(x+2)(x+2)$
B. $(x+3)(x+1)$
C. $(x+4)(x-1)$
D. Cannot be factored
Answer (1)
Find two numbers that multiply to 3 and add to 4.
The numbers are 1 and 3.
Therefore, the factored form is ( x + 3 ) ( x + 1 ) .
The final answer is ( x + 3 ) ( x + 1 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 4 x + 3 and asked to factor it. Factoring a quadratic means finding two binomials that multiply together to give us the original quadratic. We are given four possible answers: ( x + 2 ) ( x + 2 ) , ( x + 3 ) ( x + 1 ) , ( x + 4 ) ( x − 1 ) , and Cannot be factored.
Finding the Factors To factor the quadratic expression x 2 + 4 x + 3 , we need to find two numbers that multiply to the constant term (3) and add up to the coefficient of the linear term (4).
Determining the Correct Factors Let's list the factor pairs of 3: 1 and 3. Now, let's check if these factors add up to 4: 1 + 3 = 4 . Since the factors 1 and 3 satisfy both conditions, we can write the factored form of the quadratic expression as ( x + 1 ) ( x + 3 ) .
Verification To verify our answer, we can expand the factored form ( x + 1 ) ( x + 3 ) : ( x + 1 ) ( x + 3 ) = x ( x + 3 ) + 1 ( x + 3 ) = x 2 + 3 x + x + 3 = x 2 + 4 x + 3 This matches the original quadratic expression, so our factored form is correct.
Final Answer Comparing our factored form ( x + 3 ) ( x + 1 ) with the given options, we see that it matches option ( x + 3 ) ( x + 1 ) . Therefore, the correct factorization of x 2 + 4 x + 3 is ( x + 3 ) ( x + 1 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you want to build a rectangular garden with an area of x 2 + 4 x + 3 square feet. By factoring this expression into ( x + 1 ) ( x + 3 ) , you determine that the dimensions of the garden could be ( x + 1 ) feet and ( x + 3 ) feet. This allows you to plan the layout of your garden based on the available space and desired area. Factoring helps in optimizing dimensions and resource allocation in various practical scenarios.
