Question

In the following exercises, factor each trinomial of the form $x^2+bx+c$.

$a^2+14a+33$

Short answer

The factored form of the polynomial is $(a+11)(a+3)$.

Step-by-step solution

Step 1. Given Information

We are given a polynomial,

$a^2+14a+33$

Step 2. Factorizing the polynomial 

Using the product of $ac$method in factorizing the polynomial of form $a{x}^2+bx+c$, we get

$$\begin{gathered} 11\times 3=33 \\ 11+3=14 \end{gathered}$$

So, Splitting the middle term,

localid="1644849982369" $a^2+14a+33=a^2+11a+3a+33$

Grouping terms, we get

localid="1644849962131" $a^2+14a+33=\left(a^2+11a\right)+\left(3a+33\right)$

Factoring out GCF from groups, we get

localid="1661956442984" $a^2+11a+3a+33=a(a+11)+3(a+11)$

Again, factoring out GCF, we get

localid="1644850003709" $a^2+14a+33=(a+11)(a+3)$

Step 3. Checking the factorization

Multiplying the factors,

$$\begin{gathered} (a+11)(a+3)=a^2+14a+33 \\ a(a+3)+11(a+3)=a^2+14a+33 \\ {a}^2+3a+11a+33=a^2+14a+33 \\ {a}^2+14a+33=a^2+14a+33 \\ LHS=RHS \end{gathered}$$

Hence the factorization is correct.