Answer :
Let's start factorizing each given expression step-by-step.
Factorize [tex]2x^3 - 6x[/tex]:
First, identify the greatest common factor (GCF) for the terms:
[tex]\text{GCF} = 2x[/tex]
Factor out [tex]2x[/tex]:
[tex]2x(x^2 - 3)[/tex]
Factorize [tex]12y^2 - 5y[/tex]:
The GCF here is [tex]y[/tex].
Factor out [tex]y[/tex]:
[tex]y(12y - 5)[/tex]
Factorize [tex]4x^4 - 6x^3 + 2x^2 + 2x[/tex]:
Group the terms and factor by grouping:
First group: [tex]4x^4 - 6x^3[/tex], GCF is [tex]2x^3[/tex]
[tex]2x^3(2x - 3)[/tex]
Second group: [tex]2x^2 + 2x[/tex], GCF is [tex]2x[/tex]
[tex]2x(x + 1)[/tex]
Combine the factored groups:
[tex]2x(2x^3 - 3x^2 + x + 1)[/tex]
Factorize [tex]3x(x + 2) - 4(x + 2)[/tex]:
Notice the common term [tex](x + 2)[/tex], factor that out:
[tex](x + 2)(3x - 4)[/tex]
Factorize [tex]15x^2y^3 + 10xy^2[/tex]:
The GCF for both terms is [tex]5xy^2[/tex].
Factor it out:
[tex]5xy^2(3xy + 2)[/tex]
Factorize [tex]3x^3 + 27x^2 + 9x[/tex]:
The GCF is [tex]3x[/tex].
Factor it out:
[tex]3x(x^2 + 9x + 3)[/tex]
Factorize [tex]28x^4y^3 - 42x^3y^5[/tex]:
The GCF for both terms is [tex]14x^3y^3[/tex].
Factor it out:
[tex]14x^3y^3(2x - 3y^2)[/tex]
Factorize [tex]9x^3 + 27x^2 - 18x[/tex]:
The GCF is [tex]9x[/tex].
Factor it out:
[tex]9x(x^2 + 3x - 2)[/tex]
Factorize [tex]16x^3y - 32x^2y^2 + 24xy^3 - 16xy[/tex]:
The GCF here is [tex]8xy[/tex].
Factor it out:
[tex]8xy(2x^2 - 4xy + 3y^2 - 2)[/tex]
Factorize [tex]7x^5 - 28x^4 + 35x^3 - 14x^2[/tex]:
The GCF is [tex]7x^2[/tex].
Factor it out:
[tex]7x^2(x^3 - 4x^2 + 5x - 2)[/tex]
Each problem involves identifying the greatest common factor and using it to simplify the expression as far as possible.
