Q. 154E - Intermediate Algebra

Question

In the following exercises, factor each expression using any method ${\left(x+2\right)}^2-25\left(x + 2\right)-54$

Step-by-Step Solution

Answer

The factors are $\left(x-25\right)\left(x+4\right)$

Step 1. Given Information

The trinomial is ${\left(x+2\right)}^2-25\left(x + 2\right)-54$

Find factors using substitution method

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(x+2\right)}^2-25\left(x + 2\right)-54$

Substitute $x+2=u$

So, trinomial becomes $u^2-25u-54$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$$\begin{gathered} u^2-25u-54 \\ {u}^2-27u+2u-54 \end{gathered}$$

Factor it

$\left(u-27\right)\left(u+2\right)$

Step 4. Substitute back u

Substitute back $u=x+2$

$$\begin{gathered} \left(x+2-27\right)\left(x+2+2\right) \\ \left(x-25\right)\left(x+4\right) \end{gathered}$$