To factor the polynomial [tex]\(6x^4 + 15x^3y^2 + 3x^2y^3\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
The GCF of the coefficients [tex]\(6, 15,\)[/tex] and [tex]\(3\)[/tex] is [tex]\(3\)[/tex]. For the variable part, the GCF for [tex]\(x^4, x^3,\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex].
2. Factor out the GCF:
[tex]\[6x^4 + 15x^3y^2 + 3x^2y^3 = 3x^2(2x^2) + 3x^2(5xy^2) + 3x^2(y^3).\][/tex]
3. Rewrite the polynomial:
[tex]\[6x^4 + 15x^3y^2 + 3x^2y^3 = 3x^2(2x^2 + 5xy^2 + y^3).\][/tex]
Finally, the fully factored form of the polynomial in descending powers of [tex]\(x\)[/tex] is:
[tex]\[3x^2(2x^2 + 5xy^2 + y^3).\][/tex]
Therefore, placing the answer in the grid in the proper sequence:
[tex]\[\_ \_ \_ 3 \ x\ ^{2} (2 x^{2} \ + \ 5 x \ y^{2} \ + \ y^{3}).\][/tex]
