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Use the identity $(a + b) (a - b) = a^{2} - b^{2}$ to find the product $(2 a + 4 b) (2 a - 4 b)$ ?

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4 a^{2} - 9 b^{2}

{9 a}^{2} - 16 b^{2}

{16 a}^{2} - {25 b}^{2}

{4 a}^{2} - {16 b}^{2}

answer is D.

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Detailed Solution

Concept- Let

A = 2 a

B = 4 b

. Using the identity

(a + b) (a - b) = a^{2} - b^{2}

. This can help us solve the given question.
Finding the product of two algebraic expressions:
Consider the algebraic expression

a + b

and other algebraic expression

a - b

.
When we multiply the two expressions, we multiply each term of

a + b

to all the terms of

a - b

.
Now,

a (a) = a^{2}

a (- b) = - ab

b (a) = ba

b (- b) = - b^{2}

Hence,

(a + b) (a - b) = a^{2} - b^{2}

So, we have

(a + b) (a - b) = a^{2} - b^{2}

Using the identity to solve the above question.
So, we have

(2 a + 4 b) (2 a - 4 b) = (A + B) (A - B)

We know that

(a + b) (a - b) = a^{2} - b^{2}

Hence, we have

(A + B) (A - B) = A^{2} - B^{2}

.
Reverting to original variables, we have

(2 a + 4 b) (2 a - 4 b) = (2 a)^{2} - {(4 b)}^{2}

Now, we know that

(ab)^{n} = a^{n} b^{n}

Thus, we have

(2 a)^{2} = 2^{2} a^{2} = 4 a

and

(4 b)^{2} = 4^{2} b^{2} = 16 b^{2}

Hence, the correct answer is option 4)

{4 a}^{2} - {16 b}^{2}

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