Question

find the area of the rectangle. simplify your answer. image of rectangle with top side (5s^2 - 9s) and right side (3t^2 + t)

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 \( 5s^2 - 9s \) and the width is \( 3t^2 + t \). So we need to multiply these two binomials: \( A=(5s^2 - 9s)(3t^2 + t) \).

Step2: Apply the distributive property (FOIL method for binomials, extended for polynomials)

We use the distributive property \( (a + b)(c + d)=ac+ad+bc+bd \), where \( a = 5s^2 \), \( b=-9s \), \( c = 3t^2 \), and \( d = t \).
\[

$$\begin{align*} (5s^2 - 9s)(3t^2 + t)&=5s^2\times3t^2+5s^2\times t-9s\times3t^2-9s\times t\\ &=15s^2t^2 + 5s^2t-27st^2-9st \end{align*}$$

\]

Answer:

\( 15s^2t^2 + 5s^2t - 27st^2 - 9st \)