What is the product?

[tex](-3 s+2 t)(4 s-t)[/tex]

Multiply the first terms: ( − 3 s ) ( 4 s ) = − 12 s 2 .
Multiply the outer terms: ( − 3 s ) ( − t ) = 3 s t .
Multiply the inner terms: ( 2 t ) ( 4 s ) = 8 s t .
Multiply the last terms: ( 2 t ) ( − t ) = − 2 t 2 . Combine the terms: − 12 s 2 + 3 s t + 8 s t − 2 t 2 = − 12 s 2 + 11 s t − 2 t 2 . The product is − 12 s 2 + 11 s t − 2 t 2 ​ .

Explanation

Understanding the Problem We are asked to find the product of two binomials: ( − 3 s + 2 t ) and ( 4 s − t ) . We will use the distributive property (also known as the FOIL method) to expand the product.

Multiplying First Terms First, multiply the first terms: ( − 3 s ) ( 4 s ) = − 12 s 2 .

Multiplying Outer Terms Next, multiply the outer terms: ( − 3 s ) ( − t ) = 3 s t .

Multiplying Inner Terms Then, multiply the inner terms: ( 2 t ) ( 4 s ) = 8 s t .

Multiplying Last Terms Finally, multiply the last terms: ( 2 t ) ( − t ) = − 2 t 2 .

Combining Terms Now, combine all the terms: − 12 s 2 + 3 s t + 8 s t − 2 t 2 = − 12 s 2 + 11 s t − 2 t 2 .

Final Answer Therefore, the product of ( − 3 s + 2 t ) ( 4 s − t ) is − 12 s 2 + 11 s t − 2 t 2 .


Examples
Understanding how to multiply binomials is essential in many areas, such as calculating areas of rectangles with variable side lengths. For example, if you have a rectangular garden where the length is ( − 3 s + 2 t ) and the width is ( 4 s − t ) , the area of the garden is the product of these two binomials. Expanding this product helps you determine how the area changes as s and t vary, which can be useful for planning and optimizing the garden's layout. The result is − 12 s 2 + 11 s t − 2 t 2 .

Answered by GinnyAnswer | 2025-07-05