25(m squared plus 1) Squared - 9 (n sqaured minus 5) squared

The expression you've provided is:

\[
25 \left( m^2 + 1 \right)^2 - 9 \left( n^2 - 5 \right)^2
\]

Let's simplify it step-by-step.

### Step 1: Expand both squared terms

First, expand each of the squared binomials.

For the first part:
\[
\left( m^2 + 1 \right)^2 = m^4 + 2m^2 + 1
\]
Now, multiply by 25:
\[
25 \left( m^2 + 1 \right)^2 = 25 \left( m^4 + 2m^2 + 1 \right) = 25m^4 + 50m^2 + 25
\]

For the second part:
\[
\left( n^2 - 5 \right)^2 = n^4 - 10n^2 + 25
\]
Now, multiply by 9:
\[
9 \left( n^2 - 5 \right)^2 = 9 \left( n^4 - 10n^2 + 25 \right) = 9n^4 - 90n^2 + 225
\]

### Step 2: Combine both parts

Now substitute these into the original expression:
\[
25 \left( m^2 + 1 \right)^2 - 9 \left( n^2 - 5 \right)^2 = (25m^4 + 50m^2 + 25) - (9n^4 - 90n^2 + 225)
\]

Simplify by distributing the negative sign:
\[
= 25m^4 + 50m^2 + 25 - 9n^4 + 90n^2 - 225
\]

Now, combine like terms:
\[
= 25m^4 - 9n^4 + 50m^2 + 90n^2 + 25 - 225
\]
\[
= 25m^4 - 9n^4 + 50m^2 + 90n^2 - 200
\]

### Final Simplified Expression:
\[
25m^4 - 9n^4 + 50m^2 + 90n^2 - 200
\]

Would you like further clarification or have any specific questions on the steps? Here are some related questions to deepen your understanding:

1. How do you factor expressions like this if needed?
2. What would happen if we substituted specific values for \( m \) and \( n \)?
3. How would the result change if the numbers 25 and 9 were swapped in the original expression?
4. Can this expression be factored further, and if so, how?
5. How do we handle higher powers of \( m \) and \( n \) in algebraic simplifications?

**Tip:** Always expand binomials before trying to combine or simplify terms. This helps avoid mistakes and makes the problem easier to manage.

Learn how to simplify the expression 25(m^2 + 1)^2 - 9(n^2 - 5)^2 with a detailed, step-by-step guide. This solution covers binomial expansion, distribution, and simplifying polynomials, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation