Problem 37 - College Algebra Essentials

Question

Find each product. $$\left(4 x^{2}+5 x\right)\left(4 x^{2}-5 x\right)$$

Step-by-Step Solution

Answer

The product of \((4x^2 + 5x)\) and \((4x^2 - 5x)\) is \(16x^4 - 25x^2\).

Step 1: Identify the binomial expressions to multiply

We are given two binomial expressions to multiply: \((4x^2 + 5x)\) and \((4x^2 - 5x)\). We will use the FOIL method to multiply these.

Step 2: Apply the FOIL method

The first term is \(4x^2 \times 4x^2 = 16x^4\). These are the First terms in each binomial. Next, multiply the Outer terms. The outer terms are \(4x^2\) (from the first binomial) and \(-5x\) (from the second binomial). \(4x^2 \times (-5x) = -20x^3\).Then you'd multiply the Inner terms. The inner terms are \(5x\) (from the first binomial) and \(4x^2\) (from the second binomial). \(5x \times 4x^2 = 20x^3\).Last, you'd multiply the Last terms in each binomial. \(5x\) in the first binomial and \(-5x\) in the second binomial. So, \(5x \times (-5x) = -25x^2\).

Step 3: Sum up all the results

Now we have to add all of these terms that we have found together. \(16x^4 - 20x^3 + 20x^3 - 25x^2 = 16x^4 - 25x^2\). The middle terms have cancelled out as that is -20x^3 + 20x^3 = 0.