Certainly! Let's go through each part step-by-step.
### Part a: Expand [tex]\((3x + 4)^2\)[/tex]
To expand [tex]\((3x + 4)^2\)[/tex], you can use the formula for the square of a binomial: [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].
In this case, [tex]\(a = 3x\)[/tex] and [tex]\(b = 4\)[/tex].
So, we have:
[tex]\[
(3x + 4)^2 = (3x)^2 + 2 \cdot (3x) \cdot 4 + 4^2
\][/tex]
Now, calculate each term:
- [tex]\((3x)^2 = 9x^2\)[/tex]
- [tex]\(2 \cdot (3x) \cdot 4 = 2 \cdot 3 \cdot 4 \cdot x = 24x\)[/tex]
- [tex]\(4^2 = 16\)[/tex]
Putting it all together:
[tex]\[
(3x + 4)^2 = 9x^2 + 24x + 16
\][/tex]
### Part b: Expand [tex]\((2x + 4)(2x - 6)\)[/tex]
To expand [tex]\((2x + 4)(2x - 6)\)[/tex], you can use the distributive property (also known as the FOIL method for binomials).
[tex]\[
(2x + 4)(2x - 6) = 2x(2x) + 2x(-6) + 4(2x) + 4(-6)
\][/tex]
Now, calculate each term:
- [tex]\(2x \cdot 2x = 4x^2\)[/tex]
- [tex]\(2x \cdot (-6) = -12x\)[/tex]
- [tex]\(4 \cdot 2x = 8x\)[/tex]
- [tex]\(4 \cdot (-6) = -24\)[/tex]
Now, combine like terms:
[tex]\[
(2x + 4)(2x - 6) = 4x^2 - 12x + 8x - 24
\][/tex]
Combining [tex]\(-12x\)[/tex] and [tex]\(8x\)[/tex]:
[tex]\[
4x^2 - 12x + 8x - 24 = 4x^2 - 4x - 24
\][/tex]
### Final Answer:
a) [tex]\((3x + 4)^2 = 9x^2 + 24x + 16\)[/tex]
b) [tex]\((2x + 4)(2x - 6) = 4x^2 - 4x - 24\)[/tex]
