Answer:
[tex]\textsf{1.} \quad (x+3)(x+2)[/tex]
[tex]\textsf{2.} \quad (3x+1)(3x+2)(9x^2+9x-16)[/tex]
Step-by-step explanation:
Question 1
[tex]\textsf{Given expression}: \quad(x+1)^2+3(x+1)+2[/tex]
[tex]\textsf{Let }u=(x+1) \implies u^2+3u+2[/tex]
[tex]\textsf{To factor }\:\:u^2+3u+2:[/tex]
Rewrite the middle term as u + 2u:
[tex]\implies u^2+u+2u+2[/tex]
Factorize the first two terms and the last two terms separately:
[tex]\implies u(u+1)+2(u+1)[/tex]
Factor out the common term (u+1):
[tex]\implies (u+2)(u+1)[/tex]
Replace [tex]u[/tex] with [tex](x+1)[/tex] :
[tex]\implies (x+1+2)(x+1+1)[/tex]
Simplify:
[tex]\implies (x+3)(x+2)[/tex]
Question 2
[tex]\textsf{Given expression}: \quad (9x^2+9x-4)(9x^2+9x-10)-72[/tex]
Expand:
[tex]\implies 9x^2(9x^2+9x-10)+9x(9x^2+9x-10)-4(9x^2+9x-10)-72[/tex]
[tex]\implies 81x^4+81x^3-90x^2+81x^3+81x^2-90x-36x^2-36x+40-72[/tex]
Collect like terms:
[tex]\implies 81x^4+81x^3+81x^3-90x^2+81x^2-36x^2-90x-36x+40-72[/tex]
Combine like terms:
[tex]\implies 81x^4+162x^3-45x^2-126x-32[/tex]
Use the Factor Theorem:
If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).
[tex]\begin{aligned}\implies f \left(-\dfrac{1}{3}\right) & =81\left(-\dfrac{1}{3}\right)^4+162\left(-\dfrac{1}{3}\right)^3-45\left(-\dfrac{1}{3}\right)^2-126\left(-\dfrac{1}{3}\right)-32\\ & = 1-6-5+42-32\\ & = 0\end{alilgned}[/tex]
Therefore (3x + 1) is a factor.
[tex]\begin{aligned}\implies f \left(-\dfrac{2}{3}\right) & =81\left(-\dfrac{2}{3}\right)^4+162\left(-\dfrac{2}{3}\right)^3-45\left(-\dfrac{2}{3}\right)^2-126\left(-\dfrac{2}{3}\right)-32\\ & = 16-48-20+84-32\\ & = 0\end{alilgned}[/tex]
Therefore (3x + 2) is a factor.
Therefore:
[tex]\implies f(x)=(3x+1)(3x+2)(ax^2+bx+c)[/tex]
Compare the coefficient of x⁴ and the constant to find a and c:
[tex]\implies 3 \cdot 3 \cdot a=81 \implies a=9[/tex]
[tex]\implies 2c=-32 \implies c=-16[/tex]
Therefore:
[tex]\implies f(x)=(3x+1)(3x+2)(9x^2+bx-16)[/tex]
Expand:
[tex]\implies f(x)=81x^4+(81+9b)x^3-(126-9b)x^2-(144-2b)x-32[/tex]
Compare the coefficient of x³ to find b:
[tex]\implies 81+9b=162 \implies b=9[/tex]
Therefore, the fully factorized expression is:
[tex]\implies (3x+1)(3x+2)(9x^2+9x-16)[/tex]
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