1. **Factorize the expression (a):** Given: $4a^2 + 12ab$ Step 1: Identify common factors. Step 2: Both terms have a common factor of $4a$. Step 3: Factor out $4a$: $$4a^2 + 12ab = 4a(a + 3b)$$ 2. **Factorize the expression (b):** Given: $3pr + 6qr - 2pt - 4qt$ Step 1: Group terms: $(3pr + 6qr) - (2pt + 4qt)$ Step 2: Factor each group: $$3r(p + 2q) - 2t(p + 2q)$$ Step 3: Factor out the common binomial: $$ (p + 2q)(3r - 2t) $$ 3. **Inverse proportionality problem:** Given $y$ is inversely proportional to the cube root of $x$, so $y = \frac{k}{\sqrt[3]{x}}$ for some constant $k$. Step 1: Use the given values $y = 3.5$ when $x = 27$: $$3.5 = \frac{k}{\sqrt[3]{27}} = \frac{k}{3}$$ Step 2: Solve for $k$: $$k = 3.5 \times 3 = 10.5$$ Step 3: Find $y$ when $x = 125$: $$y = \frac{10.5}{\sqrt[3]{125}} = \frac{10.5}{5} = 2.1$$ 4. **Find the midpoint of $AB$:** Given points: $$A(3, -3), \, B(-8, 10)$$ Step 1: Midpoint formula is: $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ Step 2: Calculate: $$x_m = \frac{3 + (-8)}{2} = \frac{-5}{2} = -2.5$$ $$y_m = \frac{-3 + 10}{2} = \frac{7}{2} = 3.5$$ Step 3: Midpoint is: $$M(-2.5, 3.5)$$ 5. **Calculate smallest possible height of the triangle:** Given: - Area $= 15$ m$^2$ (correct to nearest $1$ m$^2$) - Base $= 6.6$ m (correct to nearest $0.1$ m) Step 1: Since area is rounded to the nearest m$^2$, smallest possible area before rounding up is: $$15 - 0.5 = 14.5\text{ m}^2$$ Step 2: Since base is rounded to nearest 0.1 m, smallest possible base before rounding up is: $$6.6 - 0.05 = 6.55\text{ m}$$ Step 3: Area formula for triangle: $$A = \frac{1}{2} \times base \times height$$ Step 4: Solve for height: $$height = \frac{2A}{base} = \frac{2 \times 14.5}{6.55} \approx 4.427$$ Step 5: The smallest possible height is approximately $4.427$ m.