Question

1. what is the area of the following rectangle?\\(2x - 6\\)\\(x^2 + 1\\)

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by the product of its length and width, i.e., \( A=\text{length} \times \text{width} \). Here, the length is \( 2x - 6 \) and the width is \( x^{2}+1 \). So we need to multiply these two expressions: \( A=(2x - 6)(x^{2}+1) \).

Step2: Expand the product using the distributive property (FOIL method for binomial and trinomial)

We distribute each term in the first binomial to each term in the second polynomial:
\[

$$\begin{align*} (2x - 6)(x^{2}+1)&=2x\times x^{2}+2x\times1-6\times x^{2}-6\times1\\ &=2x^{3}+2x - 6x^{2}-6\\ &=2x^{3}-6x^{2}+2x - 6 \end{align*}$$

\]

Answer:

The area of the rectangle is \( 2x^{3}-6x^{2}+2x - 6 \)