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Answer :
We start with the expression
[tex]$$-36x^7 + 81x^3 + 45x^3y.$$[/tex]
Step 1. Factor out the common variable factor
Notice that each term contains at least [tex]$x^3$[/tex]. Factoring [tex]$x^3$[/tex] out of every term, we have
[tex]$$
-36x^7 + 81x^3 + 45x^3y = x^3\Bigl(-36x^4 + 81 + 45y\Bigr).
$$[/tex]
Step 2. Factor out the numerical common factor
Looking at the coefficients inside the parentheses [tex]$(-36x^4 + 81 + 45y)$[/tex], we see that each of the numerical coefficients [tex]$-36$[/tex], [tex]$81$[/tex], and [tex]$45$[/tex] is divisible by [tex]$9$[/tex]. Thus, we factor [tex]$9$[/tex] out of the parentheses:
[tex]$$
-36x^4 + 81 + 45y = 9\Bigl(-4x^4 + 9 + 5y\Bigr).
$$[/tex]
Step 3. Write the fully factored form
Substitute back into the expression where we factored out [tex]$x^3$[/tex]:
[tex]$$
-36x^7 + 81x^3 + 45x^3y = 9x^3\Bigl(-4x^4 + 5y + 9\Bigr).
$$[/tex]
Thus, the fully factored expression is
[tex]$$
9x^3 \left( -4x^4 + 5y + 9 \right).
$$[/tex]
[tex]$$-36x^7 + 81x^3 + 45x^3y.$$[/tex]
Step 1. Factor out the common variable factor
Notice that each term contains at least [tex]$x^3$[/tex]. Factoring [tex]$x^3$[/tex] out of every term, we have
[tex]$$
-36x^7 + 81x^3 + 45x^3y = x^3\Bigl(-36x^4 + 81 + 45y\Bigr).
$$[/tex]
Step 2. Factor out the numerical common factor
Looking at the coefficients inside the parentheses [tex]$(-36x^4 + 81 + 45y)$[/tex], we see that each of the numerical coefficients [tex]$-36$[/tex], [tex]$81$[/tex], and [tex]$45$[/tex] is divisible by [tex]$9$[/tex]. Thus, we factor [tex]$9$[/tex] out of the parentheses:
[tex]$$
-36x^4 + 81 + 45y = 9\Bigl(-4x^4 + 9 + 5y\Bigr).
$$[/tex]
Step 3. Write the fully factored form
Substitute back into the expression where we factored out [tex]$x^3$[/tex]:
[tex]$$
-36x^7 + 81x^3 + 45x^3y = 9x^3\Bigl(-4x^4 + 5y + 9\Bigr).
$$[/tex]
Thus, the fully factored expression is
[tex]$$
9x^3 \left( -4x^4 + 5y + 9 \right).
$$[/tex]
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