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# 5.(a) $$Add: p(p – q), q(q – r) and r(r – p)$$ (b) $$Add: 2x(z – x – y) and 2y(z – y – x)$$ (c) $$Subtract: 3l(l – 4m + 5n) from 4l(10n – 3m + 2l)$$ (d) $$Subtract: 3a(a + b + c) – 2b(a – b + c) from 4c(-a + b + c)$$

(a) $$p(p – q) + q(q – r) + r(r – p)$$

=$$(p \times p) – (p \times q) + (q \times q) – (q \times r) + (r \times r) – (r \times p)$$[Opening the brackets]

= $$p^2 – pq + q^2 – qr + r^2 – rp$$

=$$p^2 + q^2 + r^2 – pq – qr – rp$$

(b) $$2x(z – x – y) + 2y(z – y – x)$$

= $$(2x \times z) – (2x \times x) – (2x \times y) + (2y \times z) – (2y \times y) – (2y \times x)$$[Opening the brackets]

= $$2xz – 2x^2 – 2xy + 2yz – 2y^2 – 2xy$$

$$= -2x^2 – 2y^2 + 2xz + 2yz – 4xy$$

$$= -2x^2 – 2y^2 – 4xy + 2yz + 2xz$$

(c) $$4l(10n – 3m + 2l) – 3l(l – 4m + 5n)$$

$$= (4l \times 10n) – (4l \times 3m) + (4l \times 2l) – (3l \times l) – (3l \times -4m) – (3l ×\times 5n)$$[Opening the brackets]

$$= 40ln – 12lm + 8l^2 – 3l^2 + 12lm – 15ln$$

$$= (40ln – 15ln) + (-12lm + 12lm) + (8l^2 – 3l^2)$$

$$= 25ln + 0 + 5l^2$$

$$= 25ln + 5l^2$$

$$= 5l^2 + 25ln$$

(d)$$[4c(-a + b + c)] – [3a(a + b + c) – 2b(a – b + c)]$$

$$= (-4ac + 4bc + 4c^2) – (3a^2 + 3ab + 3ac – 2ab + 2b^2 – 2bc)$$[Opening the brackets]

$$= -4ac + 4bc + 4c^2 – 3a^2 – 3ab – 3ac + 2ab – 2b^2 + 2bc$$

$$= -3a^2 – 2b^2 + 4c^2 – ab + 6bc – 7ac$$