Answer:
[tex]p(x) = x^{3} + 3x^{2} -4 = (x-1)(x+2)(x+2)[/tex]
Step-by-step explanation:
The given polynomial [tex]p(x) = x^{3} + 3x^{2} -4[/tex]
Now, given that (x-1) is a factor of the above equation.
Now, divide the given polynomial with the factor (x-1)
By Long division, we get
Quotient = [tex]4x^{2} + 4x + 4[/tex] and Remainder = 0
So, by the Remainder theorem
[tex]p(x) = x^{3} + 3x^{2} -4 = (4x^{2} + 4x + 4) \times (x-1)[/tex]
Now, Simplifying the quotient further, we get
[tex]4x^{2} + 4x + 4[/tex] = [tex]4x^{2} + 2x + 2x+ 4[/tex]
= [tex]x(x+2)+ 2(x+2)[/tex]
or, [tex]4x^{2} + 4x + 4[/tex] = (x+2)(x+2)
Hence, the given polynomial [tex]p(x) = x^{3} + 3x^{2} -4[/tex] can be written as a product of linear factors.
[tex]p(x) = x^{3} + 3x^{2} -4 = (x-1)(x+2)(x+2)[/tex]
