What will be the value of a and b if a^3 + y^5 + 2x^2(y^3+a)+by^3 becomes complete square.

To find the values of 'a' and 'b' that make the expression a^3 + y^5 + 2x^2(y^3+a) + by^3 a perfect square, we need to rewrite the expression in the form of a square.

Let's break down the given expression step by step:

a^3 + y^5 + 2x^2(y^3 + a) + by^3

= a^3 + y^5 + 2x^2y^3 + 2x^2a + by^3

Now, let's try to manipulate this expression to make it a perfect square. We'll focus on the terms involving 'a' and 'b' first.

Take the term '2x^2a' and rewrite it as 2xy^2a.

So, the expression now becomes:

a^3 + y^5 + 2x^2y^3 + 2xy^2a + by^3

To make a perfect square, we need to add and subtract a term that enables us to factorize a perfect square. For this case, we will consider adding and subtracting (xy^2)^2.

Adding and subtracting (xy^2)^2 to the expression:

a^3 + y^5 + 2x^2y^3 + 2xy^2a + by^3 + (xy^2)^2 - (xy^2)^2

= (a^3 + 2xy^2a + (xy^2)^2) + y^5 + 2x^2y^3 + by^3 - (xy^2)^2

= (a + xy^2)^2 + y^5 + 2x^2y^3 + by^3 - x^2y^4

Now the expression can be factored as the square of a binomial:

(a + xy^2)^2 + (y^5 + 2x^2y^3 + by^3 - x^2y^4)

Now, if the expression (y^5 + 2x^2y^3 + by^3 - x^2y^4) is equal to zero, then the given expression will be a perfect square.

Comparing with the general form of a perfect square: (m + n)^2, we can see that m = a + xy^2.

So, the value of 'a' will be -(xy^2).

Similarly, for the expression (y^5 + 2x^2y^3 + by^3 - x^2y^4), to make the expression equal to zero, the value of 'b' can be determined by setting the coefficients of all the terms to zero.

By comparing the coefficients of y^5, y^3, and y^4 terms to zero, we can set the values of 'b', 'x', and 'y'.

Hence, the values of 'a' and 'b' depend on the values of 'x' and 'y', which need to be specified to find the exact values of 'a' and 'b'.