Question

Express the given quantity in terms of the indicated variable.

The sum of the squares of two consecutive even integers; n = first integer of the two.

Short answer

The integer is $2n^2+2n+1$.

Step-by-step solution

Step 1. Concept used.

Consecutive integers differ by 1. The numbers 3 and 4 are examples of two consecutive integers.

Step 2. Add the expression.

Here, squares the expression for each integer. Then add the results

Use this fact to model the first and second integers.

$n^2,{\left(n+1\right)}^2$

So, add the expressions:

$n^2+{\left(n+1\right)}^2$

Step 3. Evaluate the expression.

Apply property${\left(a+b\right)}^2=a^2+b^2+2ab$ to expand ${\left(n+1\right)}^2$:

$=n^2+\left(n^2+1^2+2\cdot n\cdot 1\right)$

Simplify:

$=n^2+n^2+1+2n$

Add the similar term:

$=2n^2+2n+1$