Explanation:
a product of factors is only 0, if at least one of the factors is 0.
as 0 × something = 0 (always).
so, finding the zeros of
r(x) = (x - 7)² × (x² + 7)
this polynomial is of degree 4 (ultimately this will multiply x² by x² = x⁴ as highest exponent for x) and has therefore 4 zeros.
the first term is
(x - 7)²
the zero here counts twice (due to the square). in reality the curve will touch the x-axis there and not intercept it.
when will it be 0 ?
at x = 7, as 7 - 7 = 0.
so,
x = 7 are the first 2 zeroes.
now for the other term
(x² + 7)
when will that be 0 ?
when x² = -7
and that means x1 = +sqrt(7)×i
x2 = - sqrt(7)×i
these 2 zeroes are complex (or imaginary) numbers, and you cannot find them on the curve on a grid for real numbers.
but they exist and are zeroes.