Find the factor(s) of trinomial: ${x^2} + 6x + 5$

Hint:Trinomial is a polynomial expression which has exactly three terms. Factors are numbers or algebraic expressions that divide another number or algebraic expression completely without leaving any remainder. So we find the degree of the expression and use factorization to find the factors.

Complete solution step by step:
Firstly we write down the expression given in the question-
${x^2} + 6x + 5$

Now we want to find the factors of this expression so we use Factorization to reduce the expression to its factors and also keep in mind that factors of an algebraic expression are irreducible meaning that it cannot be reduced further.

We know that when an expression which has its highest power is 2, then this expression is quadratic so to find its factors we factorize it as products of its factors. We can use grouping method here to find the factors-

Grouping method: In this method we work upon the expression which is of this form ${x^2} + ax +
b$and then look for two factors $p\,{\text{and}}\,q$such that
$p \times q = b\,{\text{and}}\,p + q = a$

Then the given expression can be written as
${x^2} + (a + b)x + ab$

Now multiplying $x$inside the bracket and then taking common terms together
$
{x^2} + ax + bx + ab \\
\Rightarrow x(x + a) + b(x + a) \\
\Rightarrow (x + a)(x + b) \\
$
These two separate expressions are the factors of the given algebraic expression.
So by observing the given expression carefully we can say that we can have $p = 5\,{\text{and}}\,q =
1$such that
$
5 \times 1 = 5 = b \\
5 + 1 = 6 = a \\
$
Using the grouping method we have
\[
{x^2} + 6x + 5 \\
\Rightarrow {x^2} + (5 + 1)x + 5 \times 1 \\
\Rightarrow {x^2} + 5x + x + 5 \\
\]
Taking common factors of first two and last two elements we have
$x(x + 5) + 1(x + 5) = (x + 1)(x + 5)$

We have two factors- $(x + 1)\,{\text{and}}\,(x + 5)$ as our answer.

Note: We can always use grouping method when the given algebraic expression is not reducible to standard algebraic identities like- ${x^2} + 2xy + {y^2} = {(x + y)^2}$. After careful observation only, a hit and trial method of selecting two numbers such as $p\,{\text{and}}\,q$ should be applied to solve the problem.