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

Last updated date: 18th Sep 2024
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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,
$\Rightarrow \left( {2x + 3} \right)\left( {2x + 5} \right) = 4{x^2} + 10x + 6x + 15$
$\therefore \left( {2x + 3} \right)\left( {2x + 5} \right) = 4{x^2} + 16x + 15$
Hence, the expansion of $\left( {2x + 3} \right)\left( {2x + 5} \right)$ is $4{x^2} + 16x + 15$.