Multiply.

$\left(2 x^2+4 x-3\right)\left(x^2-2 x+5\right)$

Multiply ( 2 x 2 + 4 x − 3 ) by ( x 2 − 2 x + 5 ) .
Distribute each term: 2 x 2 ( x 2 − 2 x + 5 ) + 4 x ( x 2 − 2 x + 5 ) − 3 ( x 2 − 2 x + 5 ) .
Expand and combine like terms: 2 x 4 − 4 x 3 + 10 x 2 + 4 x 3 − 8 x 2 + 20 x − 3 x 2 + 6 x − 15 .
Simplify to get the final result: 2 x 4 − x 2 + 26 x − 15 ​ .

Explanation

Understanding the Problem We need to multiply the two polynomials ( 2 x 2 + 4 x − 3 ) and ( x 2 − 2 x + 5 ) . This involves distributing each term of the first polynomial across each term of the second polynomial and then combining like terms.

Expanding the Expression Let's multiply the polynomials step by step:


( 2 x 2 + 4 x − 3 ) ( x 2 − 2 x + 5 ) = 2 x 2 ( x 2 − 2 x + 5 ) + 4 x ( x 2 − 2 x + 5 ) − 3 ( x 2 − 2 x + 5 )

Distributing the Terms Now, distribute each term:

2 x 2 ( x 2 − 2 x + 5 ) = 2 x 4 − 4 x 3 + 10 x 2 4 x ( x 2 − 2 x + 5 ) = 4 x 3 − 8 x 2 + 20 x − 3 ( x 2 − 2 x + 5 ) = − 3 x 2 + 6 x − 15

Combining Like Terms Combine the results:

( 2 x 4 − 4 x 3 + 10 x 2 ) + ( 4 x 3 − 8 x 2 + 20 x ) + ( − 3 x 2 + 6 x − 15 ) = 2 x 4 + ( − 4 x 3 + 4 x 3 ) + ( 10 x 2 − 8 x 2 − 3 x 2 ) + ( 20 x + 6 x ) − 15
= 2 x 4 + 0 x 3 + ( 10 − 8 − 3 ) x 2 + 26 x − 15 = 2 x 4 − x 2 + 26 x − 15

Final Answer The result of the multiplication is 2 x 4 − x 2 + 26 x − 15 . Comparing this with the given options, we find that it matches the third option.

Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomials to model the load and stress distribution. Multiplying these polynomials helps them understand the combined effect of different factors, ensuring the bridge's structural integrity. Similarly, in computer graphics, polynomial multiplication is used to perform transformations and rendering of images.

Answered by GinnyAnswer | 2025-07-07