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

The correct answer/option is 4.

Let the two integers be represented by the variables $x$ and $y$.

From the problem description, 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:
$x\cdot y=165$

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:
${(x-y)}^2={(x+y)}^2-4xy$

Now, let's substitute the known values into the identity:
${(x-y)}^2={26}^2-4\left(165\right)$

Calculate the square of 26:
${26}^2=676$

Calculate the product of 4 and 165:
$4\cdot 165=660$

Substitute these values back into the equation:
${(x-y)}^2=676-660$
${(x-y)}^2=16$

Taking the square root of both sides gives the difference between the two integers:
$|x-y|=\sqrt{16}=4$

Thus, the difference between the two integers is 4.