Step 1 :We are given a polynomial $P(x) = x^{3} - 343$ and a divisor $d(x) = x + 7$. We are asked to perform polynomial long division to find the quotient $Q(x)$ and the remainder $R(x)$ such that $P(x) = d(x) \cdot Q(x) + R(x)$.
Step 2 :Performing the polynomial long division, we find that the quotient is $Q(x) = x^{2} - 7x + 49$ and the remainder is $R(x) = 0$.
Step 3 :Therefore, the polynomial $P(x)$ can be expressed as $P(x) = d(x) \cdot Q(x) + R(x)$, where $Q(x) = x^{2} - 7x + 49$ and $R(x) = 0$.
Step 4 :\(\boxed{Q(x) = x^{2} - 7x + 49, R(x) = 0}\)