What should be added to the polynomial $4{x^2} + 8x$ so that it becomes a perfect square?

Hint: Here we have to add such a number to a polynomial $4{x^2} + 8x$ that it becomes a perfect square. When a polynomial is multiplied by itself then it becomes a perfect square. A polynomial $a{x^2} + bx + c$ is a perfect square if ${b^2} = 4ac$ where $a,b,c$ are numerical coefficients. The perfect square formula is given by the expression: ${(A + B)^2} = {A^2} + {B^2} + 2AB$.

Complete step by step answer:
Here in the question we have to add such a number to the polynomial $4{x^2} + 8x$ so that it becomes a perfect square. A polynomial can be defined as an expression which consists of coefficients and variables also known as intermediates. They are a sum or difference of variables and exponents. When a polynomial is multiplied by itself then it becomes a perfect square. A polynomial $a{x^2} + bx + c$ is a perfect square if ${b^2} = 4ac$ where $a,b,c$ are numerical coefficients. We have a polynomial $4{x^2} + 8x$ so we have to add a number such that it becomes a perfect square.

Compare $4{x^2} + 8x$ by the general form of a polynomial $a{x^2} + bx + c$ .
We get $a = 4$ and $b = 8$
For $4{x^2} + 8x$ to be a perfect square we must have to add a constant i.e., $c$
For $4{x^2} + 8x$ to be a perfect square there should be must exist a relation i.e., ${b^2} = 4ac$
By this relation we get a constant number $c$ that makes $4{x^2} + 8x$ a perfect square.
Putting the value $a = 4$ and $b = 8$ in ${b^2} = 4ac$. We get,
$ \Rightarrow {(8)^2} = 4 \times 4 \times c$
$ \Rightarrow 64 = 16c$

Simplifying the above equation to get the value of $c$.
$ \Rightarrow c = \dfrac{{64}}{{16}}$
$ \Rightarrow c = 4$
Hence $4$ is added to the polynomial $4{x^2} + 8x$ to make it a perfect square.
The perfect square formula is given by the expression:
${(A + B)^2} = {A^2} + {B^2} + 2AB$.
So we can check by applying the above formula.
We can write $4{x^2} + 8x + 4$ as ${(2x)^2} + 2(2x)(2) + {(2)^2}$
${(2x)^2} + 2(2x)(2) + {(2)^2} = {(2x + 2)^2}$

Therefore, $4$ should be added to the polynomial $4{x^2} + 8x$ so that it becomes a perfect square.

Note: Polynomials are generally a sum or difference of variables and exponents. If an algebraic expression consists of square root of variables, fractional power on the variables, negative powers on the variables then the algebraic expression cannot be termed as polynomials. The square root of any number is always positive. Square and square root are inverse of each other. If we multiply an algebraic expression to itself, the product obtained is the square of that expression.