Final answer:
To build and sketch each expression as a rectangle, factorize the expressions and use the side length factors to construct the rectangles. Check if the area of the rectangles matches the product of their side length factors by multiplying the factors and comparing with the original expression.
Step-by-step explanation:
To build and sketch each expression as a rectangle, we need to factorize the expressions first. Then, we can identify the side length factors and use those to construct the rectangles. Let's go through each expression:
a. 2x^2 + 7x + 6: This expression can be factored as (2x + 3)(x + 2). The side length factors are 2x + 3 and x + 2. We can construct a rectangle with dimensions of 2x + 3 and x + 2.
b. 6x^2 + 7x + 2: This expression can be factored as (2x + 1)(3x + 2). The side length factors are 2x + 1 and 3x + 2. We can construct a rectangle with dimensions of 2x + 1 and 3x + 2.
c. x^2 + 4x + 1: This expression is a perfect square trinomial and can be factored as (x + 1)(x + 1). The side length factors are x + 1 and x + 1. We can construct a square with dimensions of x + 1.
d. 2xy + 6x + y^2 + 3y: This expression cannot be factored into distinct side length factors. So, we cannot build a rectangle for this expression.
We can check if the area of the rectangles matches the product of their side length factors by multiplying the factors for each expression and comparing the result with the original expression.