Question

find the product. write the answer in standard form.
5
$(12x^2 - 12)(9x^2 + 12)$
6
$(5x^2 + 9)(2x^2 + 3x)$
7
$(10 + 6x^2)(12x^2 + 14)$
8
$(5x^2 - 10x)^2$

Explanation:

Problem 5: \((12x^2 - 12)(9x^2 + 12)\)

Step 1: Apply the distributive property (FOIL method)

Multiply the First terms: \(12x^2 \cdot 9x^2 = 108x^4\)
Multiply the Outer terms: \(12x^2 \cdot 12 = 144x^2\)
Multiply the Inner terms: \(-12 \cdot 9x^2 = -108x^2\)
Multiply the Last terms: \(-12 \cdot 12 = -144\)

Step 2: Combine like terms

Combine the \(x^2\) terms: \(144x^2 - 108x^2 = 36x^2\)

Step 3: Write the polynomial in standard form

Arrange the terms in descending order of exponents: \(108x^4 + 36x^2 - 144\)

Step 1: Apply the distributive property (FOIL method)

Multiply the First terms: \(5x^2 \cdot 2x^2 = 10x^4\)
Multiply the Outer terms: \(5x^2 \cdot 3x = 15x^3\)
Multiply the Inner terms: \(9 \cdot 2x^2 = 18x^2\)
Multiply the Last terms: \(9 \cdot 3x = 27x\)

Step 2: Write the polynomial in standard form

Arrange the terms in descending order of exponents: \(10x^4 + 15x^3 + 18x^2 + 27x\)

Step 1: Apply the distributive property (FOIL method)

Multiply the First terms: \(10 \cdot 12x^2 = 120x^2\)
Multiply the Outer terms: \(10 \cdot 14 = 140\)
Multiply the Inner terms: \(6x^2 \cdot 12x^2 = 72x^4\)
Multiply the Last terms: \(6x^2 \cdot 14 = 84x^2\)

Step 2: Combine like terms

Combine the \(x^2\) terms: \(120x^2 + 84x^2 = 204x^2\)

Step 3: Write the polynomial in standard form

Arrange the terms in descending order of exponents: \(72x^4 + 204x^2 + 140\)

Answer:

\(108x^4 + 36x^2 - 144\)

Problem 6: \((5x^2 + 9)(2x^2 + 3x)\)