Question

find the area of the rectangle below:
image of rectangle with $3x^3$ on the side and $-4x^3 + 5$ at the bottom
(1 point)
\bigcirc $12x^6 - 15x^3$
\bigcirc $-12x^9 + 15x^3$
\bigcirc $-12x^6 + 15x^3$
\bigcirc $-x^3 + 5$
\bigcirc $11x^3 - 2$

Explanation:

Step1: Recall the area formula for 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 \( -4x^{3}+5 \) and the width is \( 3x^{3} \). So we need to compute \( 3x^{3} \times (-4x^{3}+5) \).

Step2: Apply the distributive property (FOIL for binomials)

Using the distributive property \( a(b + c)=ab+ac \), where \( a = 3x^{3} \), \( b=-4x^{3} \), and \( c = 5 \), we get:
\[

$$\begin{align*} 3x^{3}\times(-4x^{3})+3x^{3}\times5&=3\times(-4)\times x^{3}\times x^{3}+15x^{3}\\ &=- 12x^{3 + 3}+15x^{3}\\ &=-12x^{6}+15x^{3} \end{align*}$$

\]

Answer:

\(-12x^{6}+15x^{3}\) (corresponding to the option \(-12x^{6}+15x^{3}\))