Factorize the numerator. We need two numbers that multiply to 3 and add to 4. These are 1 and 3.

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Question Four a) Simplify: (y²+4y+3)/(2y²+7y+5) Step 1: Factorize the numerator. We need two numbers that multiply to 3 and add to 4. These are 1 and 3. y²+4y+3 = (y+1)(y+3); Step 2: Factorize the denominator. We need two numbers that multiply to 2 × 5 = 10 and add to 7. These are 2 and 5. 2y²+7y+5 = 2y²+2y+5y+5 = 2y(y+1)+5(y+1) = (2y+5)(y+1) Step 3: Substitute the factored expressions back into the fraction and simplify by canceling common factors. ((y+1)(y+3))/((2y+5)(y+1)) = (y+3)/(2y+5) The simplified expression is (y+3)/(2y+5). b) Express as a single fraction: (2k−2t)/(15) + (k−t)/(3) Step 1: Find a common denominator, which is 15. Multiply the second fraction by 5/5. (2k−2t)/(15) + (5(k−t))/(5 × 3) = (2k−2t)/(15) + (5k−5t)/(15) Step 2: Combine the numerators over the common denominator. ((2k−2t) + (5k−5t))/(15) = (2k−2t+5k−5t)/(15) Step 3: Combine like terms in the numerator. (7k−7t)/(15) The expression as a single fraction is (7k−7t)/(15). c) Solve: (x+4)/(2) − (2x+1)/(3) = 0 Step 1: Find the least common multiple (LCM) of the denominators 2 and 3, which is 6. Multiply the entire equation by 6. 6 ((x+4)/(2)) − 6 ((2x+1)/(3)) = 6 × 0; 3(x+4) − 2(2x+1) = 0 Step 2: Distribute the numbers into the parentheses. 3x+12 − (4x+2) = 0; 3x+12−4x−2 = 0 Step 3: Combine like terms. −x+10 = 0 Step 4: Solve for x. 10 = x; The solution is x=10. d) Solve the simultaneous equations: 2x + 3y = −4 (Equation 1); x − y = 3 (Equation 2) Step 1: From Equation 2, express x in terms of y. x = y+3 (Equation 3) Step 2: Substitute Equation 3 into Equation 1. 2(y+3) + 3y = −4; 2y+6+3y = −4 Step 3: Combine like terms and solve for y. 5y+6 = −4; 5y = −4−6; 5y = −10; y = (−10)/(5); y = −2 Step 4: Substitute the value of y back into Equation 3 to find x. x = −2+3; x = 1; The solution is x=1, y=−2. e) Factorise the following expressions: i. x² + 8x + 7 Step 1: Find two numbers that multiply to 7 and add to 8. These numbers are 1 and 7. x² + 8x + 7 = (x+1)(x+7) The factored expression is (x+1)(x+7). ii. 6 − 5b − 4b² Step 1: Rearrange the expression in standard quadratic form (descending powers of b). −4b² − 5b + 6 Step 2: Factor out −1 to make the leading coefficient positive. −(4b² + 5b − 6) Step 3: Factorize the quadratic expression 4b² + 5b − 6. We look for two numbers that multiply to 4 × (−6) = −24 and add to 5. These numbers are 8 and −3. Rewrite the middle term: 4b² + 8b − 3b − 6 Factor by grouping: 4b(b+2) − 3(b+2) (4b−3)(b+2) Step 4: Combine with the factored-out −1. −(4b−3)(b+2) Alternatively, distribute the negative sign into one of the factors, for example, (3−4b)(b+2). The factored expression is (3−4b)(b+2).

Factorize the numerator. We need two numbers that multiply to 3 and add to 4. These are 1 and 3.

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