Math  /  Algebra

QuestionPerform the indicated operations and simplify the expression. (6x21)(5x2)+(x2+3)(2x)\left(6 x^{2}-1\right)\left(5 x^{2}\right)+\left(x^{2}+3\right)(2 x)

Studdy Solution

STEP 1

1. The problem involves polynomial multiplication and addition.
2. Standard algebraic rules apply, such as the distributive property and combining like terms.
3. The goal is to simplify the expression to its simplest form.

STEP 2

1. Expand the first product (6x21)(5x2)(6x^2 - 1)(5x^2).
2. Expand the second product (x2+3)(2x)(x^2 + 3)(2x).
3. Combine the results from the two expansions.
4. Simplify the final expression by combining like terms.

STEP 3

Use the distributive property to expand (6x21)(5x2)(6x^2 - 1)(5x^2).
(6x21)(5x2)=6x25x215x2 (6x^2 - 1)(5x^2) = 6x^2 \cdot 5x^2 - 1 \cdot 5x^2

STEP 4

Multiply the terms inside the parentheses.
6x25x2=30x4 6x^2 \cdot 5x^2 = 30x^4 15x2=5x2 -1 \cdot 5x^2 = -5x^2

STEP 5

Use the distributive property to expand (x2+3)(2x)(x^2 + 3)(2x).
(x2+3)(2x)=x22x+32x (x^2 + 3)(2x) = x^2 \cdot 2x + 3 \cdot 2x

STEP 6

Multiply the terms inside the parentheses.
x22x=2x3 x^2 \cdot 2x = 2x^3 32x=6x 3 \cdot 2x = 6x

STEP 7

Combine the results from the two expansions.
30x45x2+2x3+6x 30x^4 - 5x^2 + 2x^3 + 6x

STEP 8

Simplify the final expression by writing it in standard form.
30x4+2x35x2+6x 30x^4 + 2x^3 - 5x^2 + 6x
Solution: The simplified expression is: 30x4+2x35x2+6x \boxed{30x^4 + 2x^3 - 5x^2 + 6x}

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