The sum and product of two integers are 26 and 165 respectively. The difference between these two integers is

The correct option is 4.

To find the difference between the two integers, let us define them using algebraic variables. Let the two integers be represented by $x$ and $y$.

Based on the problem statement, we are given two pieces of information:
1. The sum of the two integers is 26:
$x+y=26$
2. The product of the two integers is 165:
$x·y=165$

We want to find the absolute difference between these two integers, which is:
$|x-y|$

We can use the algebraic identity that relates the sum, product, and difference of two numbers:
${(x-y)}^2={(x+y)}^2-4xy$

Substitute the given values of the sum ($x+y=26$) and the product ($xy=165$) into the identity:
${(x-y)}^2={\left(26\right)}^2-4\left(165\right)$

Calculate the squares and product:
${\left(26\right)}^2=676$
$4·165=660$

Subtract these two values to find the value of ${(x-y)}^2$:
${(x-y)}^2=676-660$
${(x-y)}^2=16$

Taking the square root of both sides to find the difference:
$|x-y|=\sqrt{16}=4$

Thus, the difference between the two integers is indeed 4.