Expand $\left( {2x + 3} \right)\left( {2x + 5} \right)$ using the appropriate identity.

Hint: For the above question we will use the identity $\left( {a + b} \right)\left( {c + d} \right) = ac + ad + bc + bd$. Now, compare the expression with $\left( {a + b} \right)\left( {c + d} \right)$ to get the values of a, b, c, and d. By using this identity in the above question, we will get the result easily, otherwise, we will face some difficulty while simplification without using the identity in these types of problems.

Complete step-by-step solution:
We have to expand the given polynomial $\left( {2x + 3} \right)\left( {2x + 5} \right)$ using identities.
We know that,
$\left( {a + b} \right)\left( {c + d} \right) = ac + ad + bc + bd$
Now, we can compare the given expression and the expression in the above equation. So, we can get values of a, b, c and d respectively by comparison and hence put the values of (a, b, c, d) calculated in the equation to get the expansion of $\left( {2x + 3} \right)\left( {2x + 5} \right)$.
Now we will use the above identity to solve the given question as follow,
$ \Rightarrow \left( {2x + 3} \right)\left( {2x + 5} \right) = 2x \times 2x + 2x \times 5 + 3 \times 2x + 3 \times 5$
Multiply the terms,
$ \Rightarrow \left( {2x + 3} \right)\left( {2x + 5} \right) = 4{x^2} + 10x + 6x + 15$
Now, add the like terms,
$\therefore \left( {2x + 3} \right)\left( {2x + 5} \right) = 4{x^2} + 16x + 15$

Hence, the expansion of $\left( {2x + 3} \right)\left( {2x + 5} \right)$ is $4{x^2} + 16x + 15$.

Note: Be careful while doing calculation as you can make mistakes and you will get the incorrect answer. Also, remember the different identities used in algebra as it makes your approach to the question very quickly in the right direction. Also remember the fact about the identity which is that identity is equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variable ) produces the same value for all the values of the variables within a certain range of validity. If an equation satisfies all the values then it will be an identity.