Q2 Find the ramainder when the is divied by \( x^{2}+x+3 \) Q3 For the equation \( 2^{2 x+1}-3(2) \) of \( x \) aro/is? Q4 find the slant a symptote of \( f(x)=\frac{x^{3}-2 x^{2}-4 x-4}{x^{2}+x-2} \)

Let's address each question one by one. ### Q2: Find the remainder when \( f(x) \) is divided by \( x^2 + x + 3 \). To find the remainder when dividing a polynomial \( f(x) \) by a quadratic polynomial \( g(x) = x^2 + x + 3 \), we can use polynomial long division or synthetic division. The remainder will be a polynomial of degree less than 2, which can be expressed in the form \( ax + b \). However, you did not provide the polynomial \( f(x) \). Please provide the polynomial you want to divide by \( x^2 + x + 3 \). ### Q3: Solve the equation \( 2^{2x+1} - 3(2) = 0 \). First, let's rewrite the equation: \[ 2^{2x+1} - 6 = 0 \] Now, we can isolate the exponential term: \[ 2^{2x+1} = 6 \] Next, we can take the logarithm of both sides. Using base 2 logarithm: \[ 2x + 1 = \log_2(6) \] Now, solve for \( x \): \[ 2x = \log_2(6) - 1 \] \[ x = \frac{\log_2(6) - 1}{2} \] This is the solution for \( x \). ### Q4: Find the slant asymptote of \( f(x) = \frac{x^3 - 2x^2 - 4x - 4}{x^2 + x - 2} \). To find the slant (or oblique) asymptote of a rational function, we perform polynomial long division of the numerator by the denominator. 1. Divide \( x^3 \) by \( x^2 \) to get \( x \). 2. Multiply \( x \) by \( x^2 + x - 2 \) to get \( x^3 + x^2 - 2x \). 3. Subtract this from the original numerator: \[ (x^3 - 2x^2 - 4x - 4) - (x^3 + x^2 - 2x) = -3x^2 - 2x - 4 \] 4. Now, divide \( -3x^2 \) by \( x^2 \) to get \( -3 \). 5. Multiply \( -3 \) by \( x^2 + x - 2 \) to get \( -3x^2 - 3x + 6 \). 6. Subtract this from the previous result: \[ (-3x^2 - 2x - 4) - (-3x^2 - 3x + 6) = x - 10 \] The result of the division is: \[ f(x) = x - 3 + \frac{x - 10}{x^2 + x - 2} \] As \( x \) approaches infinity, the term \( \frac{x - 10}{x^2 + x - 2} \) approaches 0. Therefore, the slant asymptote is: \[ y = x - 3 \] ### Summary of Answers: - Q2: Please provide \( f(x) \) for division. - Q3: \( x = \frac{\log_2(6) - 1}{2} \) - Q4: The slant asymptote is \( y = x - 3 \).