Factor by grouping.
[tex]4 m+m n+20+5 n[/tex]
Answer (2)
Group the terms: ( 4 m + mn ) + ( 20 + 5 n ) .
Factor out the GCF from each group: m ( 4 + n ) + 5 ( 4 + n ) .
Factor out the common binomial factor: ( m + 5 ) ( 4 + n ) .
The factored expression is ( m + 5 ) ( 4 + n ) .
Explanation
Understanding the Problem We are asked to factor the expression 4 m + mn + 20 + 5 n by grouping. This means we want to rearrange and group terms in such a way that we can factor out a common binomial factor.
Grouping Terms First, let's group the terms: ( 4 m + mn ) + ( 20 + 5 n ) .
Factoring Each Group Now, we factor out the greatest common factor (GCF) from each group. From the first group ( 4 m + mn ) , we can factor out m , which gives us m ( 4 + n ) . From the second group ( 20 + 5 n ) , we can factor out 5 , which gives us 5 ( 4 + n ) . So we have: m ( 4 + n ) + 5 ( 4 + n ) .
Factoring out the Common Binomial Notice that ( 4 + n ) is a common binomial factor in both terms. We can factor out ( 4 + n ) from the entire expression: ( m + 5 ) ( 4 + n ) .
Final Factored Expression Therefore, the factored expression is ( m + 5 ) ( 4 + n ) .
Examples
Factoring by grouping is a useful technique in algebra that allows us to simplify complex expressions. For instance, suppose you are designing a rectangular garden where the area can be expressed as 4 m + mn + 20 + 5 n . By factoring this expression into ( m + 5 ) ( n + 4 ) , you can determine the possible dimensions of the garden. If m represents the length beyond a certain base length and n represents the width, this factorization helps you find the exact length and width needed to achieve the desired area. This method is also applicable in various fields such as engineering, economics, and computer science, where simplifying expressions is crucial for problem-solving.
The expression 4 m + mn + 20 + 5 n can be factored by grouping into ( m + 5 ) ( 4 + n ) . This involves first grouping the terms, factoring the groups separately, and then factoring out the common binomial factor. The final result simplifies the expression significantly.
;
