What is the simplified sum of 2xy(3x + 4y - 5z) and 5yz(2x - 3y) expressed as a single polynomial in x, y and z?

Introduction / Context:
This algebra question checks your ability to expand and add polynomials involving several variables. Each term 2xy(3x + 4y - 5z) and 5yz(2x - 3y) is a product that must be expanded first. After expanding, like terms are combined to obtain a simplified single polynomial in x, y and z.

Given Data / Assumptions:

• First expression: 2xy(3x + 4y - 5z).
• Second expression: 5yz(2x - 3y).
• We need their sum in simplified expanded form.

Concept / Approach:
Use the distributive law to expand each bracket. Multiply the coefficient outside the bracket with each term inside, keeping track of signs. Once both expressions are expanded, carefully collect like terms. Terms are like if they have the same combination of variables with the same exponents, for example x^2y and x^2y. Coefficients of like terms are then added algebraically.

Step-by-Step Solution:

Verification / Alternative check:
You can pick numerical values for x, y and z, for example x = 1, y = 1 and z = 1. Evaluate both original expressions and add them, and then evaluate the simplified polynomial. Both approaches will give the same numeric answer, confirming that the algebraic simplification is correct.

Why Other Options Are Wrong:

Common Pitfalls:
Students often forget that -10xyz and +10xyz cancel out or mis-handle the negative sign when distributing across -5z or -3y. Another common error is to treat xyz and y^2z as like terms, which they are not, since the exponents on y differ. Keeping careful track of each variable and exponent avoids these problems.

Final Answer:
6x^2y + 8xy^2 - 15y^2z