Step 1: Rearrange the terms of the quadratic expression into standard form $ax^2 + bx + c$. The given expression is $2 + 5d - 12d^2$. Rearranging it gives: $$ -12d^2 + 5d + 2 $$ Step 2: Factor out $-1$ to make the leading coefficient positive, or factor directly. Let's factor directly by finding two numbers that multiply to $a \times c = (-12) \times 2 = -24$ and add up to $b = 5$. The two numbers are $8$ and $-3$. Step 3: Rewrite the middle term $5d$ using these two numbers. $$ -12d^2 + 8d - 3d + 2 $$ Step 4: Group the terms and factor out common factors from each group. $$ (-12d^2 + 8d) + (-3d + 2) $$ Factor $4d$ from the first group and $-1$ from the second group: $$ 4d(-3d + 2) - 1(3d - 2) $$ To make the binomials identical, we can factor out $-4d$ from the first group instead: $$ -4d(3d - 2) - 1(3d - 2) $$ Step 5: Factor out the common binomial $(3d - 2)$. $$ (3d - 2)(-4d - 1) $$ This can also be written as $(2 - 3d)(4d + 1)$ by factoring out $-1$ from the first factor and $-1$ from the second factor, then multiplying the two $-1$s to get $+1$, or by factoring $-(12d^2 - 5d - 2)$. Let's verify the factorization: $(3d - 2)(-4d - 1) = 3d(-4d) + 3d(-1) - 2(-4d) - 2(-1)$ $= -12d^2 - 3d + 8d + 2$ $= -12d^2 + 5d + 2$ This matches the original expression. The completely factorised expression is $\boxed{(3d - 2)(-4d - 1)}$. That's 2 down. 3 left today — send the next one.