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【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/3Given that: The non-homogeneous linear recurrence \( \mathrm{{a}_{{n}}={a}_{{{n}-{1}}}+{n}^{{3}}+{3}{n}^{{2}}+{2}{n}} \)with the initial condition \( \mathrm{{a}_{{0}}={0}} \)The objective is to determine,(a). Show that \( \mathrm{{a}_{{n}}} \)must be a polynomial function of \( \mathrm{{n}} \) with degree \( \mathrm{{4}} \).(b). Determine the coefficient of this polynomial.In this problem, the characteristic polynomial equation associated with the homogeneous recurrence relation. So, use the iteration or the formula to find the require values.Explanation:Please refer to solution in this step.Step2/3Solving for (a),The recurrence relation for \( \mathrm{{a}_{{n}}} \) is non-homogeneous, with a particular solution given by the degree-4 polynomial \( \mathrm{{p}{\left({n}\right)}={a}{n}^{{4}}+{b}{n}^{{3}}+{c}{n}^{{2}}+{d}{n}+{e}} \).Since the degree of the polynomial is \( \mathrm{{4}} \), Now, need to find \( \mathrm{{5}} \) constants to fully specify \( \mathrm{{p}{\left({n}\right)}} \) and therefore \( \mathrm{{a}_{{n}}} \). Now, obtain these constants by solving for the coefficients of \( \mathrm{{p}{\left({n}\right)}} \) using the initial condition \( \mathrm{{a}_{{0}}={0}} \) and the recurrence relation \( \mathrm{{a}_{{n}}={a}_{{{n}-{1}}}+{n}^{{3}}+{3}{n}^{{2}}+{2}{n}} \).Hence, \( \mathrm{{a}_{{n}}} \)must be a polynomial function of \( \mathrm{{n}} \) with degree \( \mathrm{{4}} \).Explanation:In this step, the solution of non homogeneous part \( \mathrm{{p}{\left({n}\right)}} \) known as the particular solution.Explanation:Please refer to solution in this step.Step3/3Solving for (b),Using the recurrence relation and initial condition,\( \mathrm{{a}_{{1}}={a}_{{0}}+{1}^{{3}}+{3}{\left({1}^{{2}}\right)}+{2}{\left({1}\right)}={6}} \)\( \mathrm{{a}_{{2}}={a}_{{1}}+{2}^{{3}}+{3}{\left({2}^{{2}}\right)}+{2}{\left({2}\right)}={26}} \)\( \mathrm{{a}_{{3}}={ ... See the full answer