In which geometric representation the solution of the polynomial is \[(x + 5)(x + 7) = {x^2} + x(5 + 7) + 35\] ?

Hint: In this problem, we need to compare the four geometric representations with the given polynomial factors. The polynomial expression is defined as a mathematical equation made up of variables and coefficients that only includes the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

Complete step-by-step answer:
In this problem,
The given polynomial expression is \[(x + 5)(x + 7) = {x^2} + x(5 + 7) + 35\]
On comparing this polynomial factors with the given options of geometric representation which one is true, we can get

Here, the option (D) geometric representation is correct when compare to the given problem,
Area of whole rectangle \[ = \] Area of pink square Area of yellow rectangle Area of blue rectangle Area of green rectangle
By substitute the value of the diagram into the above equation, we can get
 \[(x + 5)(x + 7) = {x^2} + 5x + 7x + 35\]
By simplifying in further step, we have
 \[(x + 5)(x + 7) = {x^2} + x(5 + 7) + 35\]
Therefore, the option (D) is the correct answer.

So, the correct answer is “Option D”.

Note: We note that the geometric representations are separated as four shapes are calculated as the area of the whole rectangle which has an area of pink square and an area of yellow rectangle, an area of blue rectangle and an area of green rectangle. The mathematical expression consists of one or more algebraic terms, each consisting of a constant multiplied by one or more variables elevated to a non-negative integral power.