Problem 38 - College Algebra

Question

Find each product. $$\left(3 x^{2}+4 x\right)\left(3 x^{2}-4 x\right)$$

Step-by-Step Solution

Answer

The product is \(9x^{4} - 16x^{2}\)

Step 1: Expand the multiplication

We need to multiply every term in the first binomial by every term in the second binomial. This means that the first term \(3x^{2}\) is multiplied by every term in the second bracket and the same happens for the second term \(4x\). This results in four separate products.

Step 2: Apply the distributive property

So, let's write down those separate products: \( (3x^{2} \cdot 3x^{2}) - (3x^{2} \cdot 4x) + (4x \cdot 3x^{2}) - (4x \cdot 4x) \)

Step 3: Simplify each product

Now, for each product, we multiply the coefficients (the numbers) and add the exponents for \(x\) according to the law of exponents: \( 9x^{4} - 12x^{3} + 12x^{3} - 16x^{2} \)

Step 4: Combine like terms

-12x^{3} + 12x^{3} = 0x^{3} which simplifies to 0. So, the final simplified expression is: \( 9x^{4} - 16x^{2} \)