Let's simplify and expand each expression step-by-step.
### Part (a)
[tex]\[ 2(x - 3) + 3(5x + 7) \][/tex]
1. Distribute the 2 and 3:
[tex]\[ 2x - 6 + 15x + 21 \][/tex]
2. Combine like terms:
[tex]\[ 2x + 15x - 6 + 21 \][/tex]
[tex]\[ 17x + 15 \][/tex]
So, the simplified form of the expression is:
[tex]\[ 17x + 15 \][/tex]
### Part (b)
[tex]\[ (6 - x)(2x + 3) \][/tex]
1. Apply the distributive property (FOIL method):
[tex]\[ (6)(2x) + (6)(3) + (-x)(2x) + (-x)(3) \][/tex]
[tex]\[ 12x + 18 - 2x^2 - 3x \][/tex]
2. Combine like terms:
[tex]\[ -2x^2 + 12x - 3x + 18 \][/tex]
[tex]\[ -2x^2 + 9x + 18 \][/tex]
So, the simplified form of the expression is:
[tex]\[ -2x^2 + 9x + 18 \][/tex]
### Part (c)
[tex]\[ (x - 3)^2 \][/tex]
1. Recognize this as a perfect square trinomial:
[tex]\[ (x - 3)(x - 3) \][/tex]
2. Apply the distributive property (FOIL method):
[tex]\[ x^2 - 3x - 3x + 9 \][/tex]
3. Combine like terms:
[tex]\[ x^2 - 6x + 9 \][/tex]
So, the simplified form of the expression is:
[tex]\[ x^2 - 6x + 9 \][/tex]
### Part (d)
[tex]\[ (x - 3)(x + 2)(3x - 5) \][/tex]
1. First, let's expand the first two factors:
[tex]\[ (x - 3)(x + 2) \][/tex]
2. Apply the distributive property (FOIL method):
[tex]\[ x^2 + 2x - 3x - 6 \][/tex]
3. Combine like terms:
[tex]\[ x^2 - x - 6 \][/tex]
4. Now, multiply the result by the third factor:
[tex]\[ (x^2 - x - 6)(3x - 5) \][/tex]
5. Apply the distributive property:
[tex]\[ (x^2 - x - 6)(3x) + (x^2 - x - 6)(-5) \][/tex]
6. Use the distributive property on each part:
[tex]\[ 3x^3 - 3x^2 - 18x - 5x^2 + 5x + 30 \][/tex]
7. Combine like terms:
[tex]\[ 3x^3 - 3x^2 - 5x^2 - 18x + 5x + 30 \][/tex]
[tex]\[ 3x^3 - 8x^2 - 13x + 30 \][/tex]
So, the simplified form of the expression is:
[tex]\[ 3x^3 - 8x^2 - 13x + 30 \][/tex]
