Answer :
We begin with the expression
[tex]$$
x^6 + 125x^3.
$$[/tex]
Step 1. Factor Out the Greatest Common Factor
Notice that both terms have a factor of [tex]$x^3$[/tex]. Factor this out:
[tex]$$
x^6 + 125x^3 = x^3\left(x^3 + 125\right).
$$[/tex]
Step 2. Recognize a Sum of Cubes
The expression inside the parentheses is
[tex]$$
x^3 + 125,
$$[/tex]
and we can observe that [tex]$125 = 5^3$[/tex]. Thus, the expression is a sum of cubes:
[tex]$$
x^3 + 5^3.
$$[/tex]
Recall the sum of cubes factorization formula:
[tex]$$
a^3 + b^3 = (a + b)(a^2 - ab + b^2),
$$[/tex]
where here [tex]$a = x$[/tex] and [tex]$b = 5$[/tex].
Step 3. Factor the Sum of Cubes
Apply the formula:
[tex]$$
x^3 + 5^3 = (x + 5)(x^2 - 5x + 25).
$$[/tex]
Step 4. Write the Complete Factorization
Substitute the factorization from Step 3 back into the expression:
[tex]$$
x^6 + 125x^3 = x^3 \,(x + 5) \,(x^2 - 5x + 25).
$$[/tex]
Step 5. Match with the Given Options
The factored expression [tex]$x^3 (x+5)(x^2-5x+25)$[/tex] can be rewritten in an equivalent form. Notice that
[tex]$$
x^3 = x \cdot x^2,
$$[/tex]
so we can regroup the factors as:
[tex]$$
x^3 (x+5)(x^2-5x+25) = \left(x \,(x+5)\right) \left(x^2 (x^2-5x+25)\right).
$$[/tex]
The product [tex]$x(x+5)$[/tex] simplifies to [tex]$x^2+5x$[/tex], and
[tex]$$
x^2(x^2-5x+25) = x^4-5x^3+25x^2.
$$[/tex]
Thus, the factorization can be expressed as:
[tex]$$
\left(x^2+5x\right)\left(x^4-5x^3+25x^2\right).
$$[/tex]
Comparing with the multiple choice options, we see that this answer corresponds to option c.
Therefore, the correct answer is:
[tex]$$
\boxed{\left(x^2+5x\right)\left(x^4-5x^3+25x^2\right)}.
$$[/tex]
[tex]$$
x^6 + 125x^3.
$$[/tex]
Step 1. Factor Out the Greatest Common Factor
Notice that both terms have a factor of [tex]$x^3$[/tex]. Factor this out:
[tex]$$
x^6 + 125x^3 = x^3\left(x^3 + 125\right).
$$[/tex]
Step 2. Recognize a Sum of Cubes
The expression inside the parentheses is
[tex]$$
x^3 + 125,
$$[/tex]
and we can observe that [tex]$125 = 5^3$[/tex]. Thus, the expression is a sum of cubes:
[tex]$$
x^3 + 5^3.
$$[/tex]
Recall the sum of cubes factorization formula:
[tex]$$
a^3 + b^3 = (a + b)(a^2 - ab + b^2),
$$[/tex]
where here [tex]$a = x$[/tex] and [tex]$b = 5$[/tex].
Step 3. Factor the Sum of Cubes
Apply the formula:
[tex]$$
x^3 + 5^3 = (x + 5)(x^2 - 5x + 25).
$$[/tex]
Step 4. Write the Complete Factorization
Substitute the factorization from Step 3 back into the expression:
[tex]$$
x^6 + 125x^3 = x^3 \,(x + 5) \,(x^2 - 5x + 25).
$$[/tex]
Step 5. Match with the Given Options
The factored expression [tex]$x^3 (x+5)(x^2-5x+25)$[/tex] can be rewritten in an equivalent form. Notice that
[tex]$$
x^3 = x \cdot x^2,
$$[/tex]
so we can regroup the factors as:
[tex]$$
x^3 (x+5)(x^2-5x+25) = \left(x \,(x+5)\right) \left(x^2 (x^2-5x+25)\right).
$$[/tex]
The product [tex]$x(x+5)$[/tex] simplifies to [tex]$x^2+5x$[/tex], and
[tex]$$
x^2(x^2-5x+25) = x^4-5x^3+25x^2.
$$[/tex]
Thus, the factorization can be expressed as:
[tex]$$
\left(x^2+5x\right)\left(x^4-5x^3+25x^2\right).
$$[/tex]
Comparing with the multiple choice options, we see that this answer corresponds to option c.
Therefore, the correct answer is:
[tex]$$
\boxed{\left(x^2+5x\right)\left(x^4-5x^3+25x^2\right)}.
$$[/tex]
