Chapter 1.3, Problem 58E

To determine

To calculate: The simplified value of the expression $\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)$ .

Answer to Problem 58E

The value of the expression $\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)$ is $-x^4-3x^2-4$ .

Explanation of Solution

Given information:

The expression $\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)$ .

Formula used:

The special product formula of difference of squares to multiply the algebraic expressions is mathematically expressed as,

  $\left(A+B\right)\left(A-B\right)=A^2-B^2$

Special product of formula of perfect square of algebraic expressions which can be mathematically expressed as,

  ${\left(A+B\right)}^2=A^2+2AB+B^2$

Calculation:

Consider the given expression $\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)$ .

Recall the special product formula of difference of squares to multiply the algebraic expressions is mathematically expressed as,

  $\left(A+B\right)\left(A-B\right)=A^2-B^2$

Here, $A=x$ and $B=2+x^2$ .

Apply it,

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

Now, recall the special product of formula of perfect square of algebraic expressions which can be mathematically expressed as,

  ${\left(A+B\right)}^2=A^2+2AB+B^2$

Apply it to solve the expression ${\left(2+x^2\right)}^2$ ,

  $\begin{array}{c}\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)=x^2-{\left(2+x^2\right)}^2\\ =x^2-\left(2^2+2\left(2\right)\left(x^2\right)+{\left(x^2\right)}^2\right)\\ =x^2-\left(4+4x^2+x^4\right)\end{array}$

Simplify it further as,

  $\begin{array}{c}\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)=x^2-\left(4+4x^2+x^4\right)\\ =x^2-4-4x^2-x^4\\ =-x^4-3x^2-4\end{array}$

Thus, the value of the expression $\left(x+\left(2+x^2\right)\right)\left(x-\left(2+x^2\right)\right)$ is $-x^4-3x^2-4$ .