Multiply and add like terms: $(-2 y-6)(-4 y+2)$
Answer (1)
Expand the expression using the distributive property: ( − 2 y − 6 ) ( − 4 y + 2 ) = ( − 2 y ) ( − 4 y ) + ( − 2 y ) ( 2 ) + ( − 6 ) ( − 4 y ) + ( − 6 ) ( 2 ) .
Simplify each term: ( − 2 y ) ( − 4 y ) = 8 y 2 , ( − 2 y ) ( 2 ) = − 4 y , ( − 6 ) ( − 4 y ) = 24 y , and ( − 6 ) ( 2 ) = − 12 .
Combine like terms: − 4 y + 24 y = 20 y .
Write the simplified expression: 8 y 2 + 20 y − 12 .
Explanation
Understanding the Problem We are given the expression ( − 2 y − 6 ) ( − 4 y + 2 ) and asked to multiply and simplify by combining like terms.
Expanding the Expression We will use the distributive property (also known as the FOIL method) to expand the product of the two binomials: ( − 2 y − 6 ) ( − 4 y + 2 ) = ( − 2 y ) ( − 4 y ) + ( − 2 y ) ( 2 ) + ( − 6 ) ( − 4 y ) + ( − 6 ) ( 2 )
Simplifying Each Term Now, we simplify each term: ( − 2 y ) ( − 4 y ) = 8 y 2 ( − 2 y ) ( 2 ) = − 4 y ( − 6 ) ( − 4 y ) = 24 y ( − 6 ) ( 2 ) = − 12
Combining Like Terms Next, we combine the like terms (the terms with 'y'): − 4 y + 24 y = 20 y
Final Answer Finally, we write the simplified expression: 8 y 2 + 20 y − 12 So, the final answer is 8 y 2 + 20 y − 12 .
Examples
Understanding how to multiply binomials and combine like terms is a fundamental skill in algebra. It's used in many real-world applications, such as calculating areas, modeling growth, and solving optimization problems. For example, if you're designing a rectangular garden where the length is ( − 2 y − 6 ) meters and the width is ( − 4 y + 2 ) meters, multiplying these expressions gives you the area of the garden in terms of y . Simplifying the expression allows you to easily calculate the area for different values of y , helping you optimize the garden's size and layout.
