Expand ${(3x + 4y)^2}$

Hint: Here, we will take the given expression and use the identity for the sum of two terms and its whole square. Also, use the concept for the square where the term is multiplied with itself twice and simplify the expression for the resultant value.

Complete step-by-step answer:
Take the given expression: ${(3x + 4y)^2}$
Expand the above expression by using the identity –
${(a + b)^2} = {a^2} + 2ab + {b^2}$
${(3x + 4y)^2} = {(3x)^2} + 2(3x)(4y) + {(4y)^2}$
Here, the square of the term is calculated by multiplying the same term twice. Simplify the above expression placing the value for the square of the terms.
${(3x + 4y)^2} = 9{x^2} + 24xy + 16{y^2}$

Alternative method: The above example can be solved by another method, by using the method for finding the product of two binomials.
Take the given expression: ${(3x + 4y)^2}$
Square here means the number multiplied with itself twice, so the above expression can be re-written as –
${(3x + 4y)^2} = (3x + 4y)(3x + 4y)$
Now, expand the solution by multiplying the terms of the first bracket with the terms in the second bracket.
${(3x + 4y)^2} = 3x(3x + 4y) + 4y(3x + 4y)$
Multiply the terms inside the bracket with the terms inside the bracket. Remember when there is a positive term outside the bracket then the sign of the terms inside the bracket remains the same when opened.
${(3x + 4y)^2} = 9{x^2} + 12xy + 12xy + 16{y^2}$
Combine like terms in the above expression –
${(3x + 4y)^2} = 9{x^2} + 24xy + 16{y^2}$
So, the correct answer is “${(3x + 4y)^2} = 9{x^2} + 24xy + 16{y^2}$”.

Note: Be careful about the sign convention, while combining like terms. when you combine two terms with positive sign the resultant value is positive by adding the coefficients. But when you have two different signs you have to subtract the terms and give the sign of the bigger digit to the resultant value.