%5E2-7(x%5E2%2B4x%2B11)%2B7.jpeg "(x^2+4x+10)^2-7(x^2+4x+11)+7")
Simplifying the Expression: (x^2 + 4x + 10)^2 - 7(x^2 + 4x + 11) + 7
This problem involves simplifying a complex algebraic expression. Let's break it down step-by-step:
1. Recognize the Pattern:
Notice that the expression contains a repeated term: (x^2 + 4x + 10). This is a clue that we can use a substitution to simplify things.
2. Introduce a Substitution:
Let's substitute 'y' for the repeated term:
- y = (x^2 + 4x + 10)
Now our expression becomes:
- y^2 - 7(y + 1) + 7
3. Expand and Simplify:
- y^2 - 7y - 7 + 7
- y^2 - 7y
4. Substitute Back:
Replace 'y' with its original value:
- (x^2 + 4x + 10)^2 - 7(x^2 + 4x + 10)
5. Further Simplification (Optional):
You can choose to leave the answer as it is or expand it further. Expanding would give us:
- x^4 + 8x^3 + 28x^2 + 40x + 100 - 7x^2 - 28x - 70
- x^4 + 8x^3 + 21x^2 + 12x + 30
Therefore, the simplified forms of the given expression are:
- (x^2 + 4x + 10)^2 - 7(x^2 + 4x + 10)
- x^4 + 8x^3 + 21x^2 + 12x + 30
This method demonstrates how using substitution can greatly simplify complex expressions.