Let's factor the given polynomial step-by-step.
The given polynomial is:
[tex]\[ 4u^3 + 500 \][/tex]
To factor this polynomial completely, let's rewrite it in terms of simpler expressions that can be factored step-by-step.
1. Identify a common factor, if any:
In this case, we recognize that each term is divisible by 4:
[tex]\[ 4u^3 + 500 = 4(u^3 + 125) \][/tex]
2. Factor the remaining polynomial inside the parentheses:
Notice that [tex]\( u^3 + 125 \)[/tex] is a sum of cubes. The sum of cubes can be factored using the formula:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Here, [tex]\( u^3 \)[/tex] is [tex]\( u^3 \)[/tex] and [tex]\( 125 \)[/tex] is [tex]\( 5^3 \)[/tex]. So, the sum of cubes formula can be applied where [tex]\( a = u \)[/tex] and [tex]\( b = 5 \)[/tex]:
[tex]\[ u^3 + 5^3 = (u + 5)(u^2 - 5u + 25) \][/tex]
3. Substitute back into the original expression:
Substitute the factored form back into the equation:
[tex]\[ 4(u^3 + 125) = 4(u + 5)(u^2 - 5u + 25) \][/tex]
So, the completely factored form of the polynomial [tex]\( 4u^3 + 500 \)[/tex] is:
[tex]\[ 4(u + 5)(u^2 - 5u + 25) \][/tex]
Therefore, the correct answer is:
[tex]\[ 4(u+5)\left(u^2-5u+25\right) \][/tex]
