Chapter A.2, Problem 83E

Solution

To factor completely $2{\left(5x+1\right)}^2-18$ .

  $2\left[\left(5x-2\right)\left(5x+4\right)\right]$

Given:

The expression, $2{\left(5x+1\right)}^2-18$

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 find the factor completely, first expand the square and solving it by multiplying 2 then out the common factor from the equation use sum product pattern and solving it as shown below:

  $\begin{array}{l}2{\left(5x+1\right)}^2-18\\ 2\left({\left(5x\right)}^2+1^2+2\left(5x\right)\cdot 1\right)-18\\ 2\left(25x^2+1+10x\right)-18\\ 50x^2+2+20x-18\\ 50x^2+20x-16\\ 2\left(25x^2+10x-8\right)\\ 2\left(25x^2+20x-10x-8\right)\\ 2\left(5x\left(5x+4\right)-2\left(5x+4\right)\right)\\ 2\left[\left(5x-2\right)\left(5x+4\right)\right]\end{array}$

The original expression now become

  $2\left[\left(5x-2\right)\left(5x+4\right)\right]$