Chapter 4.6, Problem 44WE

To determine

To express: the given as a polynomial

Answer to Problem 44WE

The square of $n^2+2n-1$ is $\boxed{n^4+4n^3+2n^2-4n+1}$

Explanation of Solution

Given information: The square of $n^2+2n-1$

Concept Involved:

Write the given description as a mathematical expression, then distribute the trinomial as first step, then distribute each term further, then group and combine like terms to get the final answer.

Calculation: Write the given description “The square of $n^2+2n-1$ ” as mathematical expression

  ${\left(n^2+2n-1\right)}^2=\left(n^2+2n-1\right)\left(n^2+2n-1\right)$

Distribute the trinomial $\left(n^2+2n-1\right)$

  $n^2\left(n^2+2n-1\right)+2n\left(n^2+2n-1\right)-1\left(n^2+2n-1\right)$

Distribute further

  $n^4+2n^3-n^2+2n^3+4n^2-2n-n^2-2n+1$

Group like terms

  $n^4+2n^3+2n^3-n^2+4n^2-n^2-2n-2n+1$

Combine like terms

  $n^4+4n^3+2n^2-4n+1$

Conclusion:

The is square of $n^2+2n-1$ is $\boxed{n^4+4n^3+2n^2-4n+1}$