Expanding the Expression: (5x^2  4x + 6)(2x + 3)
This article will explore how to expand the given expression by applying the distributive property.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In mathematical terms, it can be represented as:
a(b + c) = ab + ac
Applying the Distributive Property
To expand (5x^2  4x + 6)(2x + 3), we can distribute each term of the first polynomial to each term of the second polynomial:

Distribute 2x:
 (2x)(5x^2) = 10x^3
 (2x)(4x) = 8x^2
 (2x)(6) = 12x

Distribute 3:
 (3)(5x^2) = 15x^2
 (3)(4x) = 12x
 (3)(6) = 18
Combining Like Terms
Now, we combine the terms with the same exponents:
10x^3 + 8x^2 + 15x^2 12x  12x + 18
The final expanded form of the expression is:
10x^3 + 23x^2  24x + 18
Conclusion
By applying the distributive property and combining like terms, we have successfully expanded the expression (5x^2  4x + 6)(2x + 3) into a simplified polynomial: 10x^3 + 23x^2  24x + 18. This process is essential in various mathematical operations, such as solving equations and simplifying expressions.