Final answer:
The expression 5x^3 + 40y^6 is factored by first taking out the greatest common factor of 5, resulting in 5(x^3 + 8y^6). The remaining expression is further factored as a sum of cubes to yield a fully factored form: 5(x + 2y^2)(x^2 - 2xy^2 + 4y^4).
Step-by-step explanation:
To factor the expression 5x^3 + 40y^6 completely, we look for the greatest common factor (GCF). In this case, both terms share a factor of 5. After factoring out the common factor 5, we get:
5(x^3 + 8y^6)
Now, let's look at the second part of the expression inside the parentheses: x^3 + 8y^6. This resembles the sum of cubes since 8y^6 can be written as (2y^2)^3. The sum of cubes can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Let's apply this formula to the expression:
a = x and b = 2y^2,
x^3 + (2y^2)^3 = (x + 2y^2)(x^2 - x(2y^2) + (2y^2)^2),
So the expression becomes:
5(x + 2y^2)(x^2 - 2xy^2 + 4y^4)
This is the fully factored form of the original expression. When placing these factors on a grid, each factor would be placed in its own cell in the order they appear from left to right.