Chapter A.2, Problem 33E

Solution

To expand the product $\left(x^2+x-3\right)\left(x^2+x+1\right)$ .

  $x^4+2x^3-x^2-2x-3$

Given:

The expression, $\left(x^2+x-3\right)\left(x^2+x+1\right)$

Concept Used:

• Like terms are the terms that contain the same variable with similar exponents on it. Constants are like terms as well.
• In any expression, terms are separated by plus or minus signs.
• Coefficients are the constants in front of the variable terms.

Calculation:

In order to expand the product first expand the brackets by multiplying $\left(x^2+x+1\right)$ by $x^2$ and $\left(x^2+x+1\right)$ by $x$ and $\left(x^2+x+1\right)$ by -3 and simplify by collecting like terms as shown below:

  $\begin{array}{c}\left(x^2+x-3\right)\left(x^2+x+1\right)=x^2\left(x^2+x+1\right)+x\left(x^2+x+1\right)-3\left(x^2+x+1\right)\\ =x^4+x^3+x^2+x^3+x^2+x-3x^2-3x-3\\ =x^4+2x^3-x^2-2x-3\end{array}$

So, the expression is $x^4+2x^3-x^2-2x-3$ .