Which shows one way to determine the factors of $x^3+4 x^2+5 x+20$ by grouping?

A. $x(x^2+4)+5(x^2+4)$
B. $x^2(x+4)+5(x+4)$
C. $x^2(x+5)+4(x+5)$
D. $x(x^2+5)+4 x(x^2+5)$

The correct way to determine the factors of the polynomial x 3 + 4 x 2 + 5 x + 20 by grouping is given by Option B: x 2 ( x + 4 ) + 5 ( x + 4 ) . This option results in the correct polynomial when expanded. Therefore, the answer is Option B.
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Answered by Anonymous | 2025-07-04

Test each option by factoring and expanding to see if it matches the original polynomial.
Option 1: x ( x 2 + 4 ) + 5 ( x 2 + 4 ) = ( x + 5 ) ( x 2 + 4 ) = x 3 + 5 x 2 + 4 x + 20 (incorrect).
Option 2: x 2 ( x + 4 ) + 5 ( x + 4 ) = ( x 2 + 5 ) ( x + 4 ) = x 3 + 4 x 2 + 5 x + 20 (correct).
Option 3: x 2 ( x + 5 ) + 4 ( x + 5 ) = ( x 2 + 4 ) ( x + 5 ) = x 3 + 5 x 2 + 4 x + 20 (incorrect).
Option 4: x ( x 2 + 5 ) + 4 x ( x 2 + 5 ) = 5 x ( x 2 + 5 ) = 5 x 3 + 25 x (incorrect).
The correct grouping is x 2 ( x + 4 ) + 5 ( x + 4 ) .
Therefore, the answer is x 2 ( x + 4 ) + 5 ( x + 4 ) ​

Explanation

Problem Analysis We are given the polynomial x 3 + 4 x 2 + 5 x + 20 and asked to find the correct grouping of terms that leads to its factorization. We will examine each option to see which one results in a factorization of the given polynomial.

Analyzing Option 1 Let's analyze the first option: x ( x 2 + 4 ) + 5 ( x 2 + 4 ) . This can be factored as ( x + 5 ) ( x 2 + 4 ) . Expanding this gives x 3 + 4 x + 5 x 2 + 20 = x 3 + 5 x 2 + 4 x + 20 , which is not the original polynomial.

Analyzing Option 2 Now let's analyze the second option: x 2 ( x + 4 ) + 5 ( x + 4 ) . This can be factored as ( x 2 + 5 ) ( x + 4 ) . Expanding this gives x 3 + 4 x 2 + 5 x + 20 , which is the original polynomial.

Analyzing Option 3 Let's analyze the third option: x 2 ( x + 5 ) + 4 ( x + 5 ) . This can be factored as ( x 2 + 4 ) ( x + 5 ) . Expanding this gives x 3 + 5 x 2 + 4 x + 20 , which is not the original polynomial.

Analyzing Option 4 Finally, let's analyze the fourth option: x ( x 2 + 5 ) + 4 x ( x 2 + 5 ) . This can be factored as ( x + 4 x ) ( x 2 + 5 ) = 5 x ( x 2 + 5 ) = 5 x 3 + 25 x , which is not the original polynomial.

Conclusion Therefore, the correct grouping is x 2 ( x + 4 ) + 5 ( x + 4 ) , which factors to ( x 2 + 5 ) ( x + 4 ) .


Examples
Factoring by grouping is a useful technique in algebra that allows us to simplify complex expressions. For example, suppose you are designing a rectangular garden and want to express its area in a simplified form. If the area can be represented by the expression x 3 + 4 x 2 + 5 x + 20 , factoring it into ( x + 4 ) ( x 2 + 5 ) helps you understand the dimensions of the garden. One side could be ( x + 4 ) units long, and the other side could relate to ( x 2 + 5 ) units, giving you a clearer picture for planning and design.

Answered by GinnyAnswer | 2025-07-04