Sure, let's evaluate the given expressions step-by-step for [tex]\( x = -5 \)[/tex].
### Expression (a): [tex]\( x + 2(x + 3) \)[/tex]
1. Start with the expression:
[tex]\[ x + 2(x + 3) \][/tex]
2. Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ -5 + 2(-5 + 3) \][/tex]
3. Simplify inside the parentheses:
[tex]\[ -5 + 2(-2) \][/tex]
4. Multiply:
[tex]\[ -5 + (-4) \][/tex]
5. Combine the terms:
[tex]\[ -9 \][/tex]
So, the value of expression (a) for [tex]\( x = -5 \)[/tex] is [tex]\(-9\)[/tex].
### Expression (b): [tex]\( 5(x + 5) + 3(2x + 1) \)[/tex]
1. Start with the expression:
[tex]\[ 5(x + 5) + 3(2x + 1) \][/tex]
2. Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ 5(-5 + 5) + 3(2(-5) + 1) \][/tex]
3. Simplify inside the parentheses:
[tex]\[ 5(0) + 3(-10 + 1) \][/tex]
4. Continue simplifying:
[tex]\[ 5 \cdot 0 + 3 \cdot (-9) \][/tex]
5. Multiply:
[tex]\[ 0 + (-27) \][/tex]
6. Combine the terms:
[tex]\[ -27 \][/tex]
So, the value of expression (b) for [tex]\( x = -5 \)[/tex] is [tex]\(-27\)[/tex].
### Expression (c): [tex]\( (5 + x)^2 \)[/tex]
1. Start with the expression:
[tex]\[ (5 + x)^2 \][/tex]
2. Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ (5 + (-5))^2 \][/tex]
3. Simplify inside the parentheses:
[tex]\[ (0)^2 \][/tex]
4. Square the term:
[tex]\[ 0 \][/tex]
So, the value of expression (c) for [tex]\( x = -5 \)[/tex] is [tex]\( 0 \)[/tex].
### Expression (d): [tex]\( -3(x^2 + 2x - 3) + 3(x^2 + 4x) \)[/tex]
1. Start with the expression:
[tex]\[ -3(x^2 + 2x - 3) + 3(x^2 + 4x) \][/tex]
2. Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ -3((-5)^2 + 2(-5) - 3) + 3((-5)^2 + 4(-5)) \][/tex]
3. Calculate the powers and products:
[tex]\[ -3(25 - 10 - 3) + 3(25 - 20) \][/tex]
4. Simplify the expressions inside the parentheses:
[tex]\[ -3(12) + 3(5) \][/tex]
5. Multiply:
[tex]\[ -36 + 15 \][/tex]
6. Combine the terms:
[tex]\[ -21 \][/tex]
So, the value of expression (d) for [tex]\( x = -5 \)[/tex] is [tex]\(-21\)[/tex].
### Summary
For [tex]\( x = -5 \)[/tex], the values of the given expressions are:
- (a) [tex]\( x + 2(x + 3) \)[/tex] = [tex]\(-9\)[/tex]
- (b) [tex]\( 5(x + 5) + 3(2x + 1) \)[/tex] = [tex]\(-27\)[/tex]
- (c) [tex]\( (5 + x)^2 \)[/tex] = [tex]\( 0 \)[/tex]
- (d) [tex]\( -3(x^2 + 2x - 3) + 3(x^2 + 4x) \)[/tex] = [tex]\(-21\)[/tex]
