Explanation:
a "zero" is a value of x that creates 0 as functional value of the polynomial.
for this we can use the fact that the product
0 × something = 0
for whatever that "something" is.
so, we write the polynomial expression as a product of factors. each factor is a small term that turns to 0 for a given value of x :
x + 1 = 0 for x = -1
(x + 1)² = 0 for x = -1 and multiplicity 2 (this zero counts as 2 zeroes)
x + 3 = 0 for x = -3
x + 5 = 0 for x = -5
and our polynomial is
f(x) = (x + 1)²(x + 3)(x + 5) =
= (x² + 2x + 1)(x + 3)(x + 5) =
= (x² + 2x + 1)(x² + 5x + 3x + 15) =
= (x² + 2x + 1)(x² + 8x + 15) =
= x⁴ + 8x³ + 15x² + 2x³ + 16x² + 30x + x² + 8x + 15 =
= x⁴ + 10x³ + 32x² + 38x + 15