Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2

Objectives: Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation . Methods: Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Findings: Many distinct patterns of integer solutions are obtained. Novelty: The main thrust is to illustrate different ways of obtaining various choices of solutions in integers to second-degree equations with four variables . Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Utilization of substitution strategy reduces the given equation to a ternary quadratic equation for which solutions can be found easily. Mathematics Subject Classification:11D09 Keywords: Homogeneous second degree with four variables, Solutions in integers, Factorization method, Linear transformation, Polynomial diophantine equation