Answer

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Hint: In the above given equation, we have to find the factors of the given expression. Since, this the given expression is in the cubic form, therefore we have to use the standard identity for cubic equations${A^3} - {B^3} = (A - B)({A^2} + AB + {B^2})$and then further manipulations in the equations are made.

Complete step-by-step answer:

We have the given expression as

${(a + b)^3} - {(a - b)^3}$ … (1)

Now, we know the standard identity

${A^3} - {B^3} = (A - B)({A^2} + AB + {B^2})$.

If we compare the equation (1) with the standard identity given above, we can observe that

$A = (a + b)$ and $B = (a - b)$.

Therefore, after substituting these values in the standard identity, we get the equation as

$ = [(a + b) - (a - b)][{(a + b)^2} + (a + b)(a - b) + {(a - b)^2}]$ … (2)

Now, we know that

${(a + b)^2} = {a^2} + 2ab + {b^2}$

and ${(a - b)^2} = {a^2} - 2ab + {b^2}$.

So, after using these identities in the equation (2), we get

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

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

$ = 2b(3{a^2} + {b^2})$

Hence, the correct solution is the option$(a){\text{ }}2b(3{a^2} + {b^{^2}})$.

Note: When we face such a time of questions, the key point is to have an adequate knowledge of various standard identities used like identities for quadratic equations, cubic equations, etc. With the help of these identities and some simple mathematical manipulations, the desired solution can be obtained.

Complete step-by-step answer:

We have the given expression as

${(a + b)^3} - {(a - b)^3}$ … (1)

Now, we know the standard identity

${A^3} - {B^3} = (A - B)({A^2} + AB + {B^2})$.

If we compare the equation (1) with the standard identity given above, we can observe that

$A = (a + b)$ and $B = (a - b)$.

Therefore, after substituting these values in the standard identity, we get the equation as

$ = [(a + b) - (a - b)][{(a + b)^2} + (a + b)(a - b) + {(a - b)^2}]$ … (2)

Now, we know that

${(a + b)^2} = {a^2} + 2ab + {b^2}$

and ${(a - b)^2} = {a^2} - 2ab + {b^2}$.

So, after using these identities in the equation (2), we get

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

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

$ = 2b(3{a^2} + {b^2})$

Hence, the correct solution is the option$(a){\text{ }}2b(3{a^2} + {b^{^2}})$.

Note: When we face such a time of questions, the key point is to have an adequate knowledge of various standard identities used like identities for quadratic equations, cubic equations, etc. With the help of these identities and some simple mathematical manipulations, the desired solution can be obtained.

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