$\left(2 x^2-x+4\right)\left(3 x^2+5 x-7\right)=$

Question

05-11-2026
Mathematics

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$\left(2 x^2-x+4\right)\left(3 x^2+5 x-7\right)=$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T20:59:15-05:00

Distribute each term of the first quadratic expression by the second quadratic expression: $(2x^2 - x + 4)(3x^2 + 5x - 7)$.
Perform individual multiplications: $6x^4 + 10x^3 - 14x^2 - 3x^3 - 5x^2 + 7x + 12x^2 + 20x - 28$.
Simplify the expression by combining like terms for each power of $x$.
The final expanded polynomial is $\boxed{6x^4 + 7x^3 - 7x^2 + 27x - 28}$.

Explanation

Understanding the Problem

To expand the product of the two quadratic expressions $(2x^2 - x + 4)$ and $(3x^2 + 5x - 7)$, we must distribute each term of the first expression by every term in the second expression. This process is based on the distributive property, which allows us to multiply a sum of terms by another sum of terms term-by-term. Our goal is to represent the final result as a single polynomial in the form $ax^4 + bx^3 + cx^2 + dx + e$.

Expanding the Product Terms

We expand the expression $(2x^2 - x + 4)(3x^2 + 5x - 7)$ by multiplying:
\[(2x^2)(3x^2 + 5x - 7) + (-x)(3x^2 + 5x - 7) + (4)(3x^2 + 5x - 7)\]
Distributing each term:

First term: $2x^2(3x^2) + 2x^2(5x) + 2x^2(-7) = 6x^4 + 10x^3 - 14x^2$
Second term: $-x(3x^2) - x(5x) - x(-7) = -3x^3 - 5x^2 + 7x$
Third term: $4(3x^2) + 4(5x) + 4(-7) = 12x^2 + 20x - 28$

Combining Like Terms and Final Result

Now, we group the terms with the same power of $x$:

$x^4$: $6x^4$
$x^3$: $10x^3 - 3x^3 = 7x^3$
$x^2$: $-14x^2 - 5x^2 + 12x^2 = (-14 - 5 + 12)x^2 = -7x^2$
$x$: $7x + 20x = 27x$
Constants: $-28$

Combining these results, we get the final polynomial:
$6x^4 + 7x^3 - 7x^2 + 27x - 28$.

Examples

Polynomial multiplication is fundamental in various fields, such as physics and engineering, where we analyze complex systems. For instance, when designing a bridge, engineers might need to calculate the area or volume of components whose dimensions are expressed as polynomials of a variable, such as the height or width of a support structure. By expanding these expressions, engineers can simplify calculations to determine the structural integrity and material requirements effectively.